KILLED
proof of input_zapPBSiZ7n.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
    (15) CompletionProof [UPPER BOUND(ID), 0 ms]
    (16) CpxTypedWeightedCompleteTrs
        (17) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) InliningProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxRNTS
        (25) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 225 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 512 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 120 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 377 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 252 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
        (50) CpxRNTS
        (51) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
        (54) CpxRNTS
        (55) IntTrsBoundProof [UPPER BOUND(ID), 25 ms]
        (56) CpxRNTS
        (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 847 ms]
        (60) CpxRNTS
        (61) IntTrsBoundProof [UPPER BOUND(ID), 713 ms]
        (62) CpxRNTS
(63) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(64) CdtProblem
    (65) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 161 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 139 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 13 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 153 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 166 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 18 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 258 ms]
    (128) CdtProblem
    (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 7 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (134) CdtProblem
    (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 264 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 246 ms]
    (148) CdtProblem
    (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 2 ms]
    (150) CdtProblem
    (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 60 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 232 ms]
    (166) CdtProblem
    (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 273 ms]
    (174) CdtProblem
    (175) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 2 ms]
    (178) CdtProblem
    (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 32 ms]
    (186) CdtProblem
    (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 24 ms]
    (192) CdtProblem
    (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 197 ms]
    (200) CdtProblem
    (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 209 ms]
    (206) CdtProblem
    (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CdtProblem
    (211) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (214) CdtProblem
    (215) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 6 ms]
    (216) CdtProblem
    (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (218) CdtProblem
    (219) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 26 ms]
    (224) CdtProblem
    (225) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) CdtProblem
    (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (228) CdtProblem
    (229) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (230) CdtProblem
    (231) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (232) CdtProblem
    (233) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (234) CdtProblem
    (235) CdtRewritingProof [BOTH BOUNDS(ID, ID), 16 ms]
    (236) CdtProblem
    (237) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (238) CdtProblem


----------------------------------------

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   min(0', y) -> 0'
   min(x, 0') -> 0'
   min(s(x), s(y)) -> s(min(x, y))
   max(0', y) -> y
   max(x, 0') -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0') -> 0'
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0
   min(x, 0) -> 0
   min(s(x), s(y)) -> s(min(x, y))
   max(0, y) -> y
   max(x, 0) -> x
   max(s(x), s(y)) -> s(max(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y))))

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(6)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   -(x, 0) -> x [1]
   -(s(x), s(y)) -> -(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(-(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

- => minus

----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

Rewrite Strategy: INNERMOST
----------------------------------------

(9) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f

Rewrite Strategy: INNERMOST
----------------------------------------

(11) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   min(v0, v1) -> null_min [0]
   max(v0, v1) -> null_max [0]
   twice(v0) -> null_twice [0]

And the following fresh constants:    null_minus, null_p, null_f, null_min, null_max, null_twice

----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]
   minus(v0, v1) -> null_minus [0]
   p(v0) -> null_p [0]
   f(v0, v1) -> null_f [0]
   min(v0, v1) -> null_min [0]
   max(v0, v1) -> null_max [0]
   twice(v0) -> null_twice [0]

The TRS has the following type information:
min :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
0 :: 0:s:null_minus:null_p:null_min:null_max:null_twice
s :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
max :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
p :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice
f :: 0:s:null_minus:null_p:null_min:null_max:null_twice -> 0:s:null_minus:null_p:null_min:null_max:null_twice -> null_f
null_minus :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_p :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_f :: null_f
null_min :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_max :: 0:s:null_minus:null_p:null_min:null_max:null_twice
null_twice :: 0:s:null_minus:null_p:null_min:null_max:null_twice

Rewrite Strategy: INNERMOST
----------------------------------------

(13) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   null_minus => 0
   null_p => 0
   null_f => 0
   null_min => 0
   null_max => 0
   null_twice => 0

----------------------------------------

(14)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 1 }-> f(minus(max(1 + x, 1 + y), min(1 + x, 1 + y)), p(twice(min(x, y)))) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

----------------------------------------

(15) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   f_2

(c) The following functions are completely defined:

   minus_2
   max_2
   min_2
   p_1
   twice_1

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

And the following fresh constants:    const

----------------------------------------

(16)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), s(y)) -> f(minus(max(s(x), s(y)), min(s(x), s(y))), p(twice(min(x, y)))) [1]
   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f
const :: f

Rewrite Strategy: INNERMOST
----------------------------------------

(17) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(18)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   min(0, y) -> 0 [1]
   min(x, 0) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   max(0, y) -> y [1]
   max(x, 0) -> x [1]
   max(s(x), s(y)) -> s(max(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(0), s(y)) -> f(minus(s(max(0, y)), s(min(0, y))), p(twice(0))) [4]
   f(s(x), s(0)) -> f(minus(s(max(x, 0)), s(min(x, 0))), p(twice(0))) [4]
   f(s(s(x')), s(s(y'))) -> f(minus(s(max(s(x'), s(y'))), s(min(s(x'), s(y')))), p(twice(s(min(x', y'))))) [4]
   minus(v0, v1) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
min :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
max :: 0:s -> 0:s -> 0:s
twice :: 0:s -> 0:s
minus :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
f :: 0:s -> 0:s -> f
const :: f

Rewrite Strategy: INNERMOST
----------------------------------------

(19) CpxTypedWeightedTrsToRntsProof (UPPER BOUND(ID))
Transformed the TRS into an over-approximating RNTS by (improved) Size Abstraction.
The constant constructors are abstracted as follows:

   0 => 0
   const => 0

----------------------------------------

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


----------------------------------------

(21) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

----------------------------------------

(22)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(x, 0), 1 + min(x, 0)), p(twice(0))) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 4 }-> f(minus(1 + max(0, y), 1 + min(0, y)), p(twice(0))) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + x', 1 + y'), 1 + min(1 + x', 1 + y')), p(twice(1 + min(x', y')))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   max(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   max(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   max(z, z') -{ 1 }-> 1 + max(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 1 }-> 1 + min(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


----------------------------------------

(23) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
----------------------------------------

(24)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0


----------------------------------------

(25) CpxRntsAnalysisOrderProof (BOTH BOUNDS(ID, ID))
Found the following analysis order by SCC decomposition:

   { twice }
   { max }
   { minus }
   { min }
   { p }
   { f }

----------------------------------------

(26)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}

----------------------------------------

(27) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(28)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}

----------------------------------------

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: twice
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z

----------------------------------------

(30)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: ?, size: O(n^1) [2*z]

----------------------------------------

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: twice
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z

----------------------------------------

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(twice(0))) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(twice(0))) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

----------------------------------------

(33) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(34)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

----------------------------------------

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: max
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

----------------------------------------

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {max}, {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: ?, size: O(n^1) [z + z']

----------------------------------------

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: max
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(38)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(minus(1 + max(0, z' - 1), 1 + min(0, z' - 1)), p(s')) :|: s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(minus(1 + max(z - 1, 0), 1 + min(z - 1, 0)), p(s'')) :|: s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(1 + max(1 + (z - 2), 1 + (z' - 2)), 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 1 }-> 1 + max(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']

----------------------------------------

(39) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(40)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']

----------------------------------------

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(42)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: ?, size: O(n^1) [z]

----------------------------------------

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 1 + z'

----------------------------------------

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

----------------------------------------

(45) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

----------------------------------------

(47) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: ?, size: O(n^1) [z]

----------------------------------------

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: min
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'

----------------------------------------

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 6 + z' }-> f(minus(1 + s2, 1 + min(0, z' - 1)), p(s')) :|: s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 7 }-> f(minus(1 + s3, 1 + min(z - 1, 0)), p(s'')) :|: s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 + z' }-> f(minus(1 + s4, 1 + min(1 + (z - 2), 1 + (z' - 2))), p(twice(1 + min(z - 2, z' - 2)))) :|: s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 1 }-> 1 + min(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

----------------------------------------

(51) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

----------------------------------------

(53) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z

----------------------------------------

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {p}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: ?, size: O(n^1) [z]

----------------------------------------

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: p
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

----------------------------------------

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 11 + s9 }-> f(s10, p(s'')) :|: s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 10 + s11 + s13 + 3*z' }-> f(s12, p(s14)) :|: s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 9 + s7 + 2*z' }-> f(s8, p(s')) :|: s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]

----------------------------------------

(57) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
----------------------------------------

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]

----------------------------------------

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

----------------------------------------

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: ?, size: O(1) [0]

----------------------------------------

(61) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

----------------------------------------

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 12 + s9 }-> f(s10, s16) :|: s16 >= 0, s16 <= s'', s9 >= 0, s9 <= z - 1, s10 >= 0, s10 <= 1 + s3, s3 >= 0, s3 <= 0 + (z - 1), s'' >= 0, s'' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 11 + s11 + s13 + 3*z' }-> f(s12, s17) :|: s17 >= 0, s17 <= s14, s11 >= 0, s11 <= 1 + (z - 2), s12 >= 0, s12 <= 1 + s4, s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), s4 >= 0, s4 <= 1 + (z' - 2) + (1 + (z - 2)), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 10 + s7 + 2*z' }-> f(s8, s15) :|: s15 >= 0, s15 <= s', s7 >= 0, s7 <= 0, s8 >= 0, s8 <= 1 + s2, s2 >= 0, s2 <= z' - 1 + 0, s' >= 0, s' <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   max(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   max(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   max(z, z') -{ 2 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z' - 1 + (z - 1), z - 1 >= 0, z' - 1 >= 0
   min(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 2 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s5 :|: s5 >= 0, s5 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  max: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: INF, size: O(1) [0]

----------------------------------------

(63) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(0, z0) -> c
   MIN(z0, 0) -> c1
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(0, z0) -> c3
   MAX(z0, 0) -> c4
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(0) -> c6
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(z0, 0) -> c8
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   P(s(z0)) -> c10
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, P_1, F_2

Compound Symbols:   c, c1, c2_1, c3, c4, c5_1, c6, c7_1, c8, c9_1, c10, c11_3, c12_3, c13_4


----------------------------------------

(65) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   P(s(z0)) -> c10
   -'(z0, 0) -> c8
   MIN(0, z0) -> c
   TWICE(0) -> c6
   MAX(z0, 0) -> c4
   MAX(0, z0) -> c3
   MIN(z0, 0) -> c1

----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), P(twice(min(z0, z1))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_4


----------------------------------------

(67) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   min(0, z0) -> 0
   min(z0, 0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   max(s(z0), s(z1)) -> s(max(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   p(s(z0)) -> z0
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   min_2, max_2, twice_1, -_2, p_1, f_2

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


----------------------------------------

(69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(s(z0), s(z1)) -> f(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1))))

----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


----------------------------------------

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))

----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c11_3


----------------------------------------

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c11_3, c12_3


----------------------------------------

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3


----------------------------------------

(77) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


----------------------------------------

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1)))) by 
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))

----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


----------------------------------------

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0))) by 
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))

----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c11(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2


----------------------------------------

(83) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c11(-'(max(s(0), s(x0)), min(s(0), s(x0))), MAX(s(0), s(x0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c11_2


----------------------------------------

(85) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
K tuples:none
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


----------------------------------------

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


----------------------------------------

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0))) by 
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))

----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(x0), s(0)) -> c11(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1


----------------------------------------

(91) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c11(-'(max(s(x0), s(0)), min(s(x0), s(0))), MAX(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c11_2


----------------------------------------

(93) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


----------------------------------------

(95) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1]

----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


----------------------------------------

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))

----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c12_3, c13_3, c13_2, c_1, c1_1


----------------------------------------

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))

----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


----------------------------------------

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1)))) by 
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))

----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


----------------------------------------

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0))) by 
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))

----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c12(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), -'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1


----------------------------------------

(105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c12(-'(max(s(0), s(x0)), min(s(0), s(x0))), MIN(s(0), s(x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c12_2


----------------------------------------

(107) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


----------------------------------------

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


----------------------------------------

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0))) by 
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))

----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(x0), s(0)) -> c12(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), -'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1


----------------------------------------

(113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c12(-'(max(s(x0), s(0)), min(s(x0), s(0))), MIN(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c12_2


----------------------------------------

(115) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


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(117) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


----------------------------------------

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

----------------------------------------

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c12_3, c13_3, c13_2, c11_3, c_1, c1_1, c3_1, c4_1


----------------------------------------

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))

----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1


----------------------------------------

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))

----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1


----------------------------------------

(125) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


----------------------------------------

(127) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


----------------------------------------

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(max(s(z0), s(z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(min(s(z0), s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


----------------------------------------

(131) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_3, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c6_1


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(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), min(s(z0), s(z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(max(s(z0), s(z1))), min(s(s(z0)), s(s(z1)))), p(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))

----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


----------------------------------------

(135) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


----------------------------------------

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by 
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
   F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1)))
   F(s(0), s(z1)) -> c13(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), TWICE(min(0, z1)))

----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(F(-(max(s(0), s(x0)), min(s(0), s(x0))), p(0)), TWICE(min(0, x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1


----------------------------------------

(139) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1


----------------------------------------

(141) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c13_2, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_1


----------------------------------------

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by 
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))

----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(F(-(max(s(x0), s(0)), min(s(x0), s(0))), p(0)), TWICE(min(x0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


----------------------------------------

(145) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


----------------------------------------

(147) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
S tuples:
   MIN(s(z0), s(z1)) -> c2(MIN(z0, z1))
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MIN_2, MAX_2, TWICE_1, -'_2, F_2

Compound Symbols:   c2_1, c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1


----------------------------------------

(149) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(z0), s(z1)) -> c2(MIN(z0, z1)) by 
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))

----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1


----------------------------------------

(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   F(s(x0), s(0)) -> c4(MIN(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(MIN(s(0), s(x0)))

----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MIN(s(0), s(z1)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1


----------------------------------------

(153) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 14 trailing tuple parts
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


----------------------------------------

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


----------------------------------------

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


----------------------------------------

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2


----------------------------------------

(161) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c11_2


----------------------------------------

(163) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


----------------------------------------

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


----------------------------------------

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1))) by F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1


----------------------------------------

(169) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c11(-'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c11_2


----------------------------------------

(171) CdtGraphSplitRhsProof (BOTH BOUNDS(ID, ID))
Split RHS of tuples not part of any SCC
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1


----------------------------------------

(173) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
We considered the (Usable) Rules:none
And the Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = 0
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
S tuples:
   MAX(s(z0), s(z1)) -> c5(MAX(z0, z1))
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   MAX_2, TWICE_1, -'_2, F_2, MIN_2

Compound Symbols:   c5_1, c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1


----------------------------------------

(175) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAX(s(z0), s(z1)) -> c5(MAX(z0, z1)) by 
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))

----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(z1)) -> c11(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(z1)) -> c11(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))), MAX(s(0), s(z1)))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1


----------------------------------------

(177) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   F(s(0), s(x0)) -> c(MAX(s(0), s(x0)))
   F(s(0), s(z0)) -> c10(MAX(s(0), s(z0)))
   F(s(x0), s(0)) -> c1(MAX(s(x0), s(0)))
   F(s(0), s(z0)) -> c8(MAX(s(0), s(z0)))

----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))), MAX(s(0), s(z0)))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))), MAX(s(z0), s(0)))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1


----------------------------------------

(179) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing tuple parts
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
S tuples:
   TWICE(s(z0)) -> c7(TWICE(z0))
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   TWICE_1, -'_2, F_2, MIN_2, MAX_2

Compound Symbols:   c7_1, c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2


----------------------------------------

(181) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TWICE(s(z0)) -> c7(TWICE(z0)) by 
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))

----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
S tuples:
   -'(s(z0), s(z1)) -> c9(-'(z0, z1))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   -'_2, F_2, MIN_2, MAX_2, TWICE_1

Compound Symbols:   c9_1, c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1


----------------------------------------

(183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c9(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))

----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(185) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(s(x0)), s(s(x1))) -> c6(MIN(s(x0), s(x1))) by 
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))

----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0)))) by F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0)))) by F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(x0)) -> c13(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(191) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(0), s(x0)) -> c13(TWICE(min(0, x0))) by none

----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(x0), s(0)) -> c13(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c13_1, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(s(x0), s(0)) -> c13(TWICE(min(x0, 0))) by none

----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1


----------------------------------------

(197) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(199) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = x_1 + x_2
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(203) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(205) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1] + x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MAX(x_1, x_2)) = x_1 + x_2
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(max(x_1, x_2)) = 0
   POL(min(x_1, x_2)) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1] + x_1

----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0))) by F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(209) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(211) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))

----------------------------------------

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MIN_2, MAX_2, TWICE_1, -'_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c2_1, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1


----------------------------------------

(213) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(s(y0)), s(s(y1))) -> c2(MIN(s(y0), s(y1))) by 
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))

----------------------------------------

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MIN(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MIN(s(s(z0)), s(s(0))))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


----------------------------------------

(215) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


----------------------------------------

(217) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(0)), TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


----------------------------------------

(219) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


----------------------------------------

(221) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))

----------------------------------------

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1)))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, MAX_2, TWICE_1, -'_2, MIN_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c5_1, c11_2, c7_1, c9_1, c13_1, c2_1


----------------------------------------

(223) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MAX(s(s(y0)), s(s(y1))) -> c5(MAX(s(y0), s(y1))) by 
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))

----------------------------------------

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), MAX(s(s(0)), s(s(z0))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), MAX(s(s(z0)), s(s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(225) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(227) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c13(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), TWICE(min(0, z1))) by F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(min(0, z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(229) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(231) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z1)) -> c13(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), TWICE(min(0, z1))) by F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
----------------------------------------

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(0), s(z0)) -> c13(F(-(s(max(0, z0)), min(s(0), s(z0))), p(0)), TWICE(min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(233) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(234)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(235) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), TWICE(min(z0, 0))) by F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
----------------------------------------

(236)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(0)), TWICE(min(z0, 0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1


----------------------------------------

(237) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing tuple parts
----------------------------------------

(238)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   max(s(z0), s(z1)) -> s(max(z0, z1))
   max(0, z0) -> z0
   max(z0, 0) -> z0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   min(0, z0) -> 0
   min(z0, 0) -> 0
   p(s(z0)) -> z0
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c13(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(min(0, z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(max(0, z0)), min(s(0), s(z0))), p(twice(0))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c11(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MAX(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c11(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MAX(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c11(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MAX(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c11(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MAX(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(s(z0))), s(s(s(z1)))) -> c12(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), -'(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), MIN(s(s(s(z0))), s(s(s(z1)))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(x0)), s(s(x1))) -> c12(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), -'(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), MIN(s(s(x0)), s(s(x1))))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(z0), s(z1)) -> c12(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), -'(max(s(z0), s(z1)), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c12(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), -'(max(s(s(z0)), s(s(z1))), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), min(s(s(x0)), s(s(x1)))), p(s(s(twice(min(x0, x1)))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c13(F(-(max(s(s(s(z0))), s(s(s(z1)))), min(s(s(s(z0))), s(s(s(z1))))), p(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(max(s(s(x0)), s(s(x1))), s(min(s(x0), s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(x0)), s(s(x1))) -> c13(F(-(s(max(s(x0), s(x1))), min(s(s(x0)), s(s(x1)))), p(twice(s(min(x0, x1))))), TWICE(min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(max(s(s(z0)), s(s(z1))), s(s(min(z0, z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(z1)) -> c13(F(-(s(max(z0, z1)), s(min(z0, z1))), p(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(0), s(z0)) -> c13(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(z1))) -> c13(F(-(s(s(max(z0, z1))), min(s(s(z0)), s(s(z1)))), p(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(0), s(z0)) -> c13(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z1)) -> c12(F(-(max(s(0), s(z1)), s(min(0, z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(0), s(z1)) -> c12(F(-(s(max(0, z1)), min(s(0), s(z1))), p(twice(0))), -'(max(s(0), s(z1)), min(s(0), s(z1))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c12(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c12(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c12(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c13(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c13(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(min(z0, 0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(z0), s(0)) -> c11(F(-(s(max(z0, 0)), min(s(z0), s(0))), p(twice(0))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(max(s(0), s(z0)), s(0)), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(max(s(z0), s(0)), s(0)), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   F(s(0), s(z0)) -> c11(F(-(s(z0), min(s(0), s(z0))), p(twice(min(0, z0)))), -'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(z0), s(0)) -> c11(F(-(s(z0), min(s(z0), s(0))), p(twice(min(z0, 0)))), -'(max(s(z0), s(0)), min(s(z0), s(0))))
   TWICE(s(s(y0))) -> c7(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c9(-'(s(y0), s(y1)))
   MIN(s(s(s(y0))), s(s(s(y1)))) -> c2(MIN(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c12(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c12(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
   MAX(s(s(s(y0))), s(s(s(y1)))) -> c5(MAX(s(s(y0)), s(s(y1))))
   F(s(s(0)), s(s(z0))) -> c11(F(-(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))), p(s(s(twice(0))))), -'(max(s(s(0)), s(s(z0))), min(s(s(0)), s(s(z0)))))
   F(s(s(z0)), s(s(0))) -> c11(F(-(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))), p(s(s(twice(0))))), -'(max(s(s(z0)), s(s(0))), min(s(s(z0)), s(s(0)))))
K tuples:
   F(s(0), s(x0)) -> c(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c1(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(0), s(x0)) -> c3(-'(max(s(0), s(x0)), min(s(0), s(x0))))
   F(s(x0), s(0)) -> c4(-'(max(s(x0), s(0)), min(s(x0), s(0))))
   F(s(s(x0)), s(s(x1))) -> c6(TWICE(min(s(x0), s(x1))))
   F(s(0), s(z0)) -> c8(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(0), s(z0)) -> c10(-'(max(s(0), s(z0)), min(s(0), s(z0))))
   F(s(s(s(y0))), s(s(s(y1)))) -> c6(MIN(s(s(y0)), s(s(y1))))
   F(s(0), s(z0)) -> c13(TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c13(TWICE(min(z0, 0)))
Defined Rule Symbols:   -_2, max_2, min_2, p_1, twice_1

Defined Pair Symbols:   F_2, TWICE_1, -'_2, MIN_2, MAX_2

Compound Symbols:   c11_3, c_1, c1_1, c12_3, c3_1, c4_1, c13_3, c6_1, c13_2, c12_2, c8_1, c10_1, c11_2, c7_1, c9_1, c13_1, c2_1, c5_1