KILLED
proof of input_EuMfmJy07j.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# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(4) CdtProblem
    (5) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxRelTRS
    (7) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 8 ms]
    (10) typed CpxTrs
    (11) OrderProof [LOWER BOUND(ID), 0 ms]
    (12) typed CpxTrs
    (13) RewriteLemmaProof [LOWER BOUND(ID), 344 ms]
    (14) BEST
        (15) proven lower bound
            (16) LowerBoundPropagationProof [FINISHED, 0 ms]
            (17) BOUNDS(n^1, INF)
        (18) typed CpxTrs
            (19) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (20) typed CpxTrs
            (21) RewriteLemmaProof [LOWER BOUND(ID), 62 ms]
            (22) typed CpxTrs
            (23) RewriteLemmaProof [LOWER BOUND(ID), 58 ms]
            (24) typed CpxTrs
            (25) RewriteLemmaProof [LOWER BOUND(ID), 51 ms]
            (26) typed CpxTrs
            (27) RewriteLemmaProof [LOWER BOUND(ID), 41 ms]
            (28) typed CpxTrs
(29) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(30) CpxTRS
(31) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(32) CdtProblem
    (33) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CpxRelTRS
        (39) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxTRS
        (41) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (42) CpxWeightedTrs
            (43) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (44) CpxWeightedTrs
            (45) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (46) CpxTypedWeightedTrs
            (47) CompletionProof [UPPER BOUND(ID), 0 ms]
            (48) CpxTypedWeightedCompleteTrs
                (49) NarrowingProof [BOTH BOUNDS(ID, ID), 21 ms]
                (50) CpxTypedWeightedCompleteTrs
                (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (52) CpxRNTS
                (53) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (54) CpxRNTS
                (55) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (56) CpxRNTS
                (57) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (58) CpxRNTS
                (59) IntTrsBoundProof [UPPER BOUND(ID), 210 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 74 ms]
                (62) CpxRNTS
                (63) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (64) CpxRNTS
                (65) IntTrsBoundProof [UPPER BOUND(ID), 221 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
                (68) CpxRNTS
                (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (70) CpxRNTS
                (71) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
                (74) CpxRNTS
                (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (76) CpxRNTS
                (77) IntTrsBoundProof [UPPER BOUND(ID), 232 ms]
                (78) CpxRNTS
                (79) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
                (80) CpxRNTS
                (81) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (82) CpxRNTS
                (83) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
                (86) CpxRNTS
                (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (88) CpxRNTS
                (89) IntTrsBoundProof [UPPER BOUND(ID), 28 ms]
                (90) CpxRNTS
                (91) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
                (92) CpxRNTS
                (93) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (94) CpxRNTS
                (95) IntTrsBoundProof [UPPER BOUND(ID), 1731 ms]
                (96) CpxRNTS
                (97) IntTrsBoundProof [UPPER BOUND(ID), 892 ms]
                (98) CpxRNTS
            (99) CompletionProof [UPPER BOUND(ID), 0 ms]
            (100) CpxTypedWeightedCompleteTrs
                (101) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (102) CpxRNTS
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 9 ms]
    (104) CdtProblem
        (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (112) CdtProblem
        (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CdtProblem
        (117) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem
        (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 10 ms]
        (122) CdtProblem
        (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (126) CdtProblem
        (127) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (130) CdtProblem
        (131) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (148) CdtProblem
        (149) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 140 ms]
        (150) CdtProblem
        (151) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms]
        (152) CdtProblem
        (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
        (156) CdtProblem
        (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (162) CdtProblem
        (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (164) CdtProblem
        (165) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (166) CdtProblem
        (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CdtProblem
        (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CdtProblem
        (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (172) CdtProblem
        (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (174) CdtProblem
        (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (176) CdtProblem
        (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (178) CdtProblem
        (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (184) CdtProblem
        (185) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 168 ms]
        (186) CdtProblem
        (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 12 ms]
        (188) CdtProblem
        (189) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (190) CdtProblem
        (191) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms]
        (192) CdtProblem
        (193) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 13 ms]
        (196) CdtProblem
        (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (198) CdtProblem
        (199) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1207 ms]
        (200) CdtProblem
        (201) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (202) CdtProblem
        (203) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 183 ms]
        (204) CdtProblem
        (205) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (206) CdtProblem
        (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (208) CdtProblem
        (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (210) CdtProblem
        (211) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1168 ms]
        (212) CdtProblem
        (213) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (214) CdtProblem
        (215) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 161 ms]
        (216) CdtProblem
        (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 9 ms]
        (218) CdtProblem
        (219) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (220) CdtProblem
        (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (222) CdtProblem
        (223) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1134 ms]
        (224) CdtProblem
        (225) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (226) CdtProblem
        (227) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (228) CdtProblem
        (229) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1107 ms]
        (230) CdtProblem
        (231) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (232) CdtProblem
(233) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(234) CpxWeightedTrs
    (235) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (236) CpxWeightedTrs
    (237) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (238) CpxTypedWeightedTrs
    (239) CompletionProof [UPPER BOUND(ID), 0 ms]
    (240) CpxTypedWeightedCompleteTrs
        (241) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (242) CpxTypedWeightedCompleteTrs
        (243) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (244) CpxRNTS
        (245) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (246) CpxRNTS
        (247) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (248) CpxRNTS
        (249) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (250) CpxRNTS
        (251) IntTrsBoundProof [UPPER BOUND(ID), 159 ms]
        (252) CpxRNTS
        (253) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
        (254) CpxRNTS
        (255) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (256) CpxRNTS
        (257) IntTrsBoundProof [UPPER BOUND(ID), 287 ms]
        (258) CpxRNTS
        (259) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
        (260) CpxRNTS
        (261) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (262) CpxRNTS
        (263) IntTrsBoundProof [UPPER BOUND(ID), 215 ms]
        (264) CpxRNTS
        (265) IntTrsBoundProof [UPPER BOUND(ID), 177 ms]
        (266) CpxRNTS
        (267) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (268) CpxRNTS
        (269) IntTrsBoundProof [UPPER BOUND(ID), 1569 ms]
        (270) CpxRNTS
        (271) IntTrsBoundProof [UPPER BOUND(ID), 932 ms]
        (272) CpxRNTS
    (273) CompletionProof [UPPER BOUND(ID), 0 ms]
    (274) CpxTypedWeightedCompleteTrs
        (275) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 6 ms]
        (276) CpxRNTS


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

(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:

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

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

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

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

(4)
Obligation:
Complexity Dependency Tuples Problem

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

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

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6_1, c7_3, c8_3, c9_3, c10_3


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

(5) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

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


The TRS R consists of the following rules:

   -'(z0, 0) -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(z0, 0) -> c2
   MIN(0, z0) -> c3
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(0) -> c5
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

The (relative) TRS S consists of the following rules:

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
   f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

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

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

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


The TRS R consists of the following rules:

   -'(z0, 0') -> c
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(z0, 0') -> c2
   MIN(0', z0) -> c3
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(0') -> c5
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

The (relative) TRS S consists of the following rules:

   -(z0, 0') -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0') -> 0'
   min(0', z0) -> 0'
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0') -> 0'
   twice(s(z0)) -> s(s(twice(z0)))
   f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
   f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

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

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

(10)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10

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

(11) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
-', MIN, TWICE, F, -, min, twice, f

They will be analysed ascendingly in the following order:
-' < F
MIN < F
TWICE < F
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(12)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
-', MIN, TWICE, F, -, min, twice, f

They will be analysed ascendingly in the following order:
-' < F
MIN < F
TWICE < F
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(13) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)

Induction Base:
-'(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1)
c

Induction Step:
-'(gen_0':s8_11(+(n13_11, 1)), gen_0':s8_11(+(n13_11, 1))) ->_R^Omega(1)
c1(-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11))) ->_IH
c1(gen_c:c17_11(c14_11))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(14)
Complex Obligation (BEST)

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

(15)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
-', MIN, TWICE, F, -, min, twice, f

They will be analysed ascendingly in the following order:
-' < F
MIN < F
TWICE < F
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(16) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(17)
BOUNDS(n^1, INF)

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

(18)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
MIN, TWICE, F, -, min, twice, f

They will be analysed ascendingly in the following order:
MIN < F
TWICE < F
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(19) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)

Induction Base:
MIN(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(1)
c2

Induction Step:
MIN(gen_0':s8_11(+(n426_11, 1)), gen_0':s8_11(+(n426_11, 1))) ->_R^Omega(1)
c4(MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11))) ->_IH
c4(gen_c2:c3:c49_11(c427_11))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(20)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
TWICE, F, -, min, twice, f

They will be analysed ascendingly in the following order:
TWICE < F
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(21) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11)

Induction Base:
TWICE(gen_0':s8_11(0)) ->_R^Omega(1)
c5

Induction Step:
TWICE(gen_0':s8_11(+(n945_11, 1))) ->_R^Omega(1)
c6(TWICE(gen_0':s8_11(n945_11))) ->_IH
c6(gen_c5:c610_11(c946_11))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(22)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)
TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
-, F, min, twice, f

They will be analysed ascendingly in the following order:
- < F
min < F
twice < F
- < f
min < f
twice < f

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

(23) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
-(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0)

Induction Base:
-(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0)
gen_0':s8_11(0)

Induction Step:
-(gen_0':s8_11(+(n1336_11, 1)), gen_0':s8_11(+(n1336_11, 1))) ->_R^Omega(0)
-(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) ->_IH
gen_0':s8_11(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(24)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)
TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11)
-(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
min, F, twice, f

They will be analysed ascendingly in the following order:
min < F
twice < F
min < f
twice < f

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

(25) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0)

Induction Base:
min(gen_0':s8_11(0), gen_0':s8_11(0)) ->_R^Omega(0)
0'

Induction Step:
min(gen_0':s8_11(+(n2044_11, 1)), gen_0':s8_11(+(n2044_11, 1))) ->_R^Omega(0)
s(min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11))) ->_IH
s(gen_0':s8_11(c2045_11))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(26)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)
TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11)
-(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0)
min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
twice, F, f

They will be analysed ascendingly in the following order:
twice < F
twice < f

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

(27) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
twice(gen_0':s8_11(n2554_11)) -> gen_0':s8_11(*(2, n2554_11)), rt in Omega(0)

Induction Base:
twice(gen_0':s8_11(0)) ->_R^Omega(0)
0'

Induction Step:
twice(gen_0':s8_11(+(n2554_11, 1))) ->_R^Omega(0)
s(s(twice(gen_0':s8_11(n2554_11)))) ->_IH
s(s(gen_0':s8_11(*(2, c2555_11))))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(28)
Obligation:
Innermost TRS:
Rules:
-'(z0, 0') -> c
-'(s(z0), s(z1)) -> c1(-'(z0, z1))
MIN(z0, 0') -> c2
MIN(0', z0) -> c3
MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
TWICE(0') -> c5
TWICE(s(z0)) -> c6(TWICE(z0))
F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
-(z0, 0') -> z0
-(s(z0), s(z1)) -> -(z0, z1)
min(z0, 0') -> 0'
min(0', z0) -> 0'
min(s(z0), s(z1)) -> s(min(z0, z1))
twice(0') -> 0'
twice(s(z0)) -> s(s(twice(z0)))
f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))

Types:
-' :: 0':s -> 0':s -> c:c1
0' :: 0':s
c :: c:c1
s :: 0':s -> 0':s
c1 :: c:c1 -> c:c1
MIN :: 0':s -> 0':s -> c2:c3:c4
c2 :: c2:c3:c4
c3 :: c2:c3:c4
c4 :: c2:c3:c4 -> c2:c3:c4
TWICE :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
F :: 0':s -> 0':s -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
- :: 0':s -> 0':s -> 0':s
min :: 0':s -> 0':s -> 0':s
twice :: 0':s -> 0':s
c8 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c:c1 -> c2:c3:c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c5:c6 -> c2:c3:c4 -> c7:c8:c9:c10
f :: 0':s -> 0':s -> f
hole_c:c11_11 :: c:c1
hole_0':s2_11 :: 0':s
hole_c2:c3:c43_11 :: c2:c3:c4
hole_c5:c64_11 :: c5:c6
hole_c7:c8:c9:c105_11 :: c7:c8:c9:c10
hole_f6_11 :: f
gen_c:c17_11 :: Nat -> c:c1
gen_0':s8_11 :: Nat -> 0':s
gen_c2:c3:c49_11 :: Nat -> c2:c3:c4
gen_c5:c610_11 :: Nat -> c5:c6
gen_c7:c8:c9:c1011_11 :: Nat -> c7:c8:c9:c10


Lemmas:
-'(gen_0':s8_11(n13_11), gen_0':s8_11(n13_11)) -> gen_c:c17_11(n13_11), rt in Omega(1 + n13_11)
MIN(gen_0':s8_11(n426_11), gen_0':s8_11(n426_11)) -> gen_c2:c3:c49_11(n426_11), rt in Omega(1 + n426_11)
TWICE(gen_0':s8_11(n945_11)) -> gen_c5:c610_11(n945_11), rt in Omega(1 + n945_11)
-(gen_0':s8_11(n1336_11), gen_0':s8_11(n1336_11)) -> gen_0':s8_11(0), rt in Omega(0)
min(gen_0':s8_11(n2044_11), gen_0':s8_11(n2044_11)) -> gen_0':s8_11(n2044_11), rt in Omega(0)
twice(gen_0':s8_11(n2554_11)) -> gen_0':s8_11(*(2, n2554_11)), rt in Omega(0)


Generator Equations:
gen_c:c17_11(0) <=> c
gen_c:c17_11(+(x, 1)) <=> c1(gen_c:c17_11(x))
gen_0':s8_11(0) <=> 0'
gen_0':s8_11(+(x, 1)) <=> s(gen_0':s8_11(x))
gen_c2:c3:c49_11(0) <=> c2
gen_c2:c3:c49_11(+(x, 1)) <=> c4(gen_c2:c3:c49_11(x))
gen_c5:c610_11(0) <=> c5
gen_c5:c610_11(+(x, 1)) <=> c6(gen_c5:c610_11(x))
gen_c7:c8:c9:c1011_11(0) <=> hole_c7:c8:c9:c105_11
gen_c7:c8:c9:c1011_11(+(x, 1)) <=> c7(gen_c7:c8:c9:c1011_11(x), c, c2)


The following defined symbols remain to be analysed:
F, f
----------------------------------------

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

(30)
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:

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

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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

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

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

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5, c6_1, c7_3, c8_3, c9_3, c10_3


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

(33) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   MIN(z0, 0) -> c2
   MIN(0, z0) -> c3
   -'(z0, 0) -> c
   TWICE(0) -> c5

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   f(s(z0), s(z1)) -> f(-(z1, min(z0, z1)), s(twice(min(z0, z1))))
   f(s(z0), s(z1)) -> f(-(z0, min(z0, z1)), s(twice(min(z0, z1))))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1, f_2

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3


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

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c8_3, c9_3, c10_3


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

(37) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(38)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))

The (relative) TRS S consists of the following rules:

   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))

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

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

(40)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))

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

(41) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(42)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   -(z0, 0) -> z0 [0]
   -(s(z0), s(z1)) -> -(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

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

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

- => minus

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

(44)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

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

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

(46)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
MIN :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
TWICE :: s:0 -> c6
c6 :: c6 -> c6
F :: s:0 -> s:0 -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
minus :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
twice :: s:0 -> s:0
c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
0 :: s:0

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

(47) 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:

   -'_2
   MIN_2
   TWICE_1
   F_2

(c) The following functions are completely defined:

   minus_2
   min_2
   twice_1

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   twice(v0) -> 0 [0]

And the following fresh constants:    const, const1, const2, const3

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

(48)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   twice(v0) -> 0 [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
MIN :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
TWICE :: s:0 -> c6
c6 :: c6 -> c6
F :: s:0 -> s:0 -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
minus :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
twice :: s:0 -> s:0
c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
0 :: s:0
const :: c1
const1 :: c4
const2 :: c6
const3 :: c7:c8:c9:c10

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

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

(50)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> 0 [0]
   min(v0, v1) -> 0 [0]
   twice(v0) -> 0 [0]

The TRS has the following type information:
-' :: s:0 -> s:0 -> c1
s :: s:0 -> s:0
c1 :: c1 -> c1
MIN :: s:0 -> s:0 -> c4
c4 :: c4 -> c4
TWICE :: s:0 -> c6
c6 :: c6 -> c6
F :: s:0 -> s:0 -> c7:c8:c9:c10
c7 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
minus :: s:0 -> s:0 -> s:0
min :: s:0 -> s:0 -> s:0
twice :: s:0 -> s:0
c8 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
c9 :: c7:c8:c9:c10 -> c1 -> c4 -> c7:c8:c9:c10
c10 :: c7:c8:c9:c10 -> c6 -> c4 -> c7:c8:c9:c10
0 :: s:0
const :: c1
const1 :: c4
const2 :: c6
const3 :: c7:c8:c9:c10

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

(51) 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
   const1 => 0
   const2 => 0
   const3 => 0

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z0, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z1, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0


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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0


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

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

   { MIN }
   { twice }
   { -' }
   { minus }
   { min }
   { TWICE }
   { F }

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F}

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

(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:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F}

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: MIN
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:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {MIN}, {twice}, {-'}, {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: ?, size: O(1) [0]

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

(61) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ 1 }-> 1 + MIN(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {-'}, {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]

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

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {twice}, {-'}, {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]

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

(65) 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

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

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

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

Function symbols to be analyzed: {-'}, {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]

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

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {-'}, {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 1 }-> 1 + -'(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {minus}, {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]

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

(77) 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

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {min}, {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]

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

(83) 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'

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(85) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z - 1, min(z - 1, z' - 1)) + s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + TWICE(min(z - 1, z' - 1)) + s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(minus(z' - 1, min(z - 1, z' - 1)), 1 + twice(min(z - 1, z' - 1))) + -'(z' - 1, min(z - 1, z' - 1)) + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + min(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  min: runtime: O(1) [0], size: O(n^1) [z']

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

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {TWICE}, {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  min: runtime: O(1) [0], size: O(n^1) [z']

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(91) 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: z

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s13, 1 + s15) + TWICE(s16) + s'' :|: s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ z' }-> 1 + F(s24, 1 + s26) + TWICE(s27) + s2 :|: s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ 1 }-> 1 + TWICE(z - 1) :|: z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  min: runtime: O(1) [0], size: O(n^1) [z']
  TWICE: runtime: O(n^1) [z], size: O(1) [0]

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

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

(94)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {F}
Previous analysis results are:
  MIN: runtime: O(n^1) [z'], size: O(1) [0]
  twice: runtime: O(1) [0], size: O(n^1) [2*z]
  -': runtime: O(n^1) [z'], size: O(1) [0]
  minus: runtime: O(1) [0], size: O(n^1) [z]
  min: runtime: O(1) [0], size: O(n^1) [z']
  TWICE: runtime: O(n^1) [z], size: O(1) [0]

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

(95) 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

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

(96)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(97) 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: ?

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

(98)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ z' }-> 1 + s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s16 + z' }-> 1 + F(s13, 1 + s15) + s30 + s'' :|: s30 >= 0, s30 <= 0, s12 >= 0, s12 <= z' - 1, s13 >= 0, s13 <= z' - 1, s14 >= 0, s14 <= z' - 1, s15 >= 0, s15 <= 2 * s14, s16 >= 0, s16 <= z' - 1, s'' >= 0, s'' <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s21 + z' }-> 1 + F(s18, 1 + s20) + s22 + s1 :|: s17 >= 0, s17 <= z' - 1, s18 >= 0, s18 <= z - 1, s19 >= 0, s19 <= z' - 1, s20 >= 0, s20 <= 2 * s19, s21 >= 0, s21 <= z' - 1, s22 >= 0, s22 <= 0, s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s27 + z' }-> 1 + F(s24, 1 + s26) + s31 + s2 :|: s31 >= 0, s31 <= 0, s23 >= 0, s23 <= z' - 1, s24 >= 0, s24 <= z - 1, s25 >= 0, s25 <= z' - 1, s26 >= 0, s26 <= 2 * s25, s27 >= 0, s27 <= z' - 1, s2 >= 0, s2 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ s10 + z' }-> 1 + F(s7, 1 + s9) + s11 + s' :|: s6 >= 0, s6 <= z' - 1, s7 >= 0, s7 <= z' - 1, s8 >= 0, s8 <= z' - 1, s9 >= 0, s9 <= 2 * s8, s10 >= 0, s10 <= z' - 1, s11 >= 0, s11 <= 0, s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   MIN(z, z') -{ z' }-> 1 + s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   TWICE(z) -{ z }-> 1 + s29 :|: s29 >= 0, s29 <= 0, z - 1 >= 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   min(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   min(z, z') -{ 0 }-> 1 + s28 :|: s28 >= 0, s28 <= z' - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= z - 1, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: z >= 0
   twice(z) -{ 0 }-> 1 + (1 + s3) :|: s3 >= 0, s3 <= 2 * (z - 1), z - 1 >= 0

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

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

(99) 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]
   min(v0, v1) -> null_min [0]
   twice(v0) -> null_twice [0]
   -'(v0, v1) -> null_-' [0]
   MIN(v0, v1) -> null_MIN [0]
   TWICE(v0) -> null_TWICE [0]
   F(v0, v1) -> null_F [0]

And the following fresh constants:    null_minus, null_min, null_twice, null_-', null_MIN, null_TWICE, null_F

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

(100)
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:

   -'(s(z0), s(z1)) -> c1(-'(z0, z1)) [1]
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) [1]
   TWICE(s(z0)) -> c6(TWICE(z0)) [1]
   F(s(z0), s(z1)) -> c7(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c8(F(minus(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c9(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) [1]
   F(s(z0), s(z1)) -> c10(F(minus(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) [1]
   minus(z0, 0) -> z0 [0]
   minus(s(z0), s(z1)) -> minus(z0, z1) [0]
   min(z0, 0) -> 0 [0]
   min(0, z0) -> 0 [0]
   min(s(z0), s(z1)) -> s(min(z0, z1)) [0]
   twice(0) -> 0 [0]
   twice(s(z0)) -> s(s(twice(z0))) [0]
   minus(v0, v1) -> null_minus [0]
   min(v0, v1) -> null_min [0]
   twice(v0) -> null_twice [0]
   -'(v0, v1) -> null_-' [0]
   MIN(v0, v1) -> null_MIN [0]
   TWICE(v0) -> null_TWICE [0]
   F(v0, v1) -> null_F [0]

The TRS has the following type information:
-' :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c1:null_-'
s :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice
c1 :: c1:null_-' -> c1:null_-'
MIN :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c4:null_MIN
c4 :: c4:null_MIN -> c4:null_MIN
TWICE :: s:0:null_minus:null_min:null_twice -> c6:null_TWICE
c6 :: c6:null_TWICE -> c6:null_TWICE
F :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> c7:c8:c9:c10:null_F
c7 :: c7:c8:c9:c10:null_F -> c1:null_-' -> c4:null_MIN -> c7:c8:c9:c10:null_F
minus :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice
min :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice
twice :: s:0:null_minus:null_min:null_twice -> s:0:null_minus:null_min:null_twice
c8 :: c7:c8:c9:c10:null_F -> c6:null_TWICE -> c4:null_MIN -> c7:c8:c9:c10:null_F
c9 :: c7:c8:c9:c10:null_F -> c1:null_-' -> c4:null_MIN -> c7:c8:c9:c10:null_F
c10 :: c7:c8:c9:c10:null_F -> c6:null_TWICE -> c4:null_MIN -> c7:c8:c9:c10:null_F
0 :: s:0:null_minus:null_min:null_twice
null_minus :: s:0:null_minus:null_min:null_twice
null_min :: s:0:null_minus:null_min:null_twice
null_twice :: s:0:null_minus:null_min:null_twice
null_-' :: c1:null_-'
null_MIN :: c4:null_MIN
null_TWICE :: c6:null_TWICE
null_F :: c7:c8:c9:c10:null_F

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

(101) 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_min => 0
   null_twice => 0
   null_-' => 0
   null_MIN => 0
   null_TWICE => 0
   null_F => 0

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

(102)
Obligation:
Complexity RNTS consisting of the following rules:

   -'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   -'(z, z') -{ 1 }-> 1 + -'(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z0, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z0, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + TWICE(min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + F(minus(z1, min(z0, z1)), 1 + twice(min(z0, z1))) + -'(z1, min(z0, z1)) + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   MIN(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   MIN(z, z') -{ 1 }-> 1 + MIN(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   TWICE(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   TWICE(z) -{ 1 }-> 1 + TWICE(z0) :|: z = 1 + z0, z0 >= 0
   min(z, z') -{ 0 }-> 0 :|: z = z0, z0 >= 0, z' = 0
   min(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   min(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   min(z, z') -{ 0 }-> 1 + min(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   minus(z, z') -{ 0 }-> minus(z0, z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   twice(z) -{ 0 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 0 }-> 1 + (1 + twice(z0)) :|: z = 1 + z0, z0 >= 0

Only complete derivations are relevant for the runtime complexity.

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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c7(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), -'(z1, min(z0, z1)), MIN(z0, z1)) by 
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))), -'(0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3


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

(105) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_3, c9_3, c10_3, c7_3, c7_1, c7_2


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c8(F(-(z1, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_3, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c9(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), -'(z0, min(z0, z1)), MIN(z0, z1)) by 
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))), -'(0, min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3


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

(113) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing tuple parts
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_3, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(z1)) -> c10(F(-(z0, min(z0, z1)), s(twice(min(z0, z1)))), TWICE(min(z0, z1)), MIN(z0, z1)) by 
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)), MIN(z0, 0))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)), MIN(0, z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3


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

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(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)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, min(z0, 0)), s(twice(0)))) by 
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_2, c7_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, min(0, z0)), s(twice(0))), -'(z0, min(0, z0))) by 
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0)))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3


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

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(min(z0, 0))))) by 
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0)))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0)))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0)))))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1


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

(131) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(x0), s(0)) -> c7(F(0, s(twice(min(x0, 0)))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_2, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(min(0, z0)))), -'(z0, min(0, z0))) by 
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2


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

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_3, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2


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

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by 
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by 
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by 
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(F(0, s(twice(min(x0, 0)))), TWICE(min(x0, 0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3


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

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
K tuples:none
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1


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

(149) 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)) -> c8(TWICE(min(x0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(min(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1]

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c8_2, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_1


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

(151) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by 
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1


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

(153) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_3, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1


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

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, min(z0, 0)), s(twice(0))), -'(z0, min(z0, 0))) by 
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3


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

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_1, c9_2, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3


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

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, min(0, z0)), s(twice(0)))) by 
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_2, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3


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

(163) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(min(z0, 0)))), -'(z0, min(z0, 0))) by 
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))), -'(0, min(0, 0)))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2


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

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0)))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c9_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(min(0, z0))))) by 
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0)))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0)))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0)))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1


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

(169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(x0)) -> c9(F(0, s(twice(min(0, x0)))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), min(s(z0), s(z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_3, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1


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

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(min(s(z0), s(z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1))) by 
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3


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

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, min(z0, 0)), s(twice(0))), TWICE(min(z0, 0))) by 
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3


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

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, min(0, z0)), s(twice(0))), TWICE(min(0, z0))) by 
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3


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

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(min(z0, 0)))), TWICE(min(z0, 0))) by 
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c10_2, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3


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

(181) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(min(0, z0)))), TWICE(min(0, z0))) by 
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(F(0, s(twice(min(0, x0)))), TWICE(min(0, x0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2


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

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1


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

(185) 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)) -> c10(TWICE(min(0, x0)))
We considered the (Usable) Rules:none
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1, x_2)) = [1]
   POL(MIN(x_1, x_2)) = 0
   POL(TWICE(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(min(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(twice(x_1)) = 0

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0))) by F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c1_1, c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1


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

(189) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
S tuples:
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), -'(s(0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1


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

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), MIN(s(z0), s(0)))
S tuples:
   MIN(s(z0), s(z1)) -> c4(MIN(z0, z1))
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), MIN(s(z0), s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c4_1, c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1


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

(193) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MIN(s(z0), s(z1)) -> c4(MIN(z0, z1)) by 
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), MIN(s(z0), s(0)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))), MIN(s(0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), MIN(s(z0), s(0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), MIN(s(0), s(z0)))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), MIN(s(0), s(z0)))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))), MIN(s(z0), s(0)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1


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

(195) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 16 trailing tuple parts
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1


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

(197) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0)))
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(0)))
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1


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

(199) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by 
   F(s(x0), s(0)) -> c7(F(0, s(0)))

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7(F(0, s(0)))
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7(F(0, s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1


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

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7


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

(203) 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)) -> c7
We considered the (Usable) Rules:none
And the Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7
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] + x_2
   POL(MIN(x_1, x_2)) = x_1 + x_2
   POL(TWICE(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(min(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1]

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7
S tuples:
   TWICE(s(z0)) -> c6(TWICE(z0))
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c7
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c6_1, c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7


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

(205) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TWICE(s(z0)) -> c6(TWICE(z0)) by 
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   F(s(x0), s(0)) -> c7
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(x0), s(0)) -> c7
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c7, c6_1


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

(207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(x0), s(0)) -> c7

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(209) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0)))
----------------------------------------

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(211) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by 
   F(s(0), s(0)) -> c7(F(0, s(0)))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7(F(0, s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7(F(0, s(0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7


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

(215) 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(0)) -> c7
We considered the (Usable) Rules:none
And the Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7
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] + x_2
   POL(MIN(x_1, x_2)) = x_1 + x_2
   POL(TWICE(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7) = 0
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(min(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(twice(x_1)) = [1]

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(0), s(0)) -> c7
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7


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

(217) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0))) by F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
----------------------------------------

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(0)) -> c7
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   F(s(0), s(0)) -> c7
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1, c7


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

(219) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c7

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(221) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(0), s(0)) -> c7(F(-(0, 0), s(0)))
----------------------------------------

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(223) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0), s(0)) -> c7(F(-(0, 0), s(0))) by 
   F(s(0), s(0)) -> c7(F(0, s(0)))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
   F(s(0), s(0)) -> c7(F(0, s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(225) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(0), s(0)) -> c7(F(0, s(0)))

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

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(227) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(twice(0)))) by F(s(z0), s(0)) -> c7(F(-(0, 0), s(0)))
----------------------------------------

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
   F(s(z0), s(0)) -> c7(F(-(0, 0), s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(229) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0), s(0)) -> c7(F(-(0, 0), s(0))) by 
   F(s(x0), s(0)) -> c7(F(0, s(0)))

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

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
   F(s(x0), s(0)) -> c7(F(0, s(0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

(231) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(s(x0), s(0)) -> c7(F(0, s(0)))

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

(232)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   min(z0, 0) -> 0
   min(0, z0) -> 0
   min(s(z0), s(z1)) -> s(min(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
Tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(0)) -> c7(F(-(0, 0), s(twice(0))))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(twice(0))), -'(z0, min(0, z0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
S tuples:
   F(s(s(x0)), s(s(x1))) -> c7(F(-(s(x1), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x1), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c7(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z1), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c7(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), -'(s(z0), min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c7(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z1)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c7(F(-(0, min(x0, 0)), s(0)))
   F(s(0), s(x0)) -> c7(F(-(x0, min(0, x0)), s(0)), -'(x0, min(0, x0)))
   F(s(0), s(z0)) -> c7(F(z0, s(twice(min(0, z0)))), -'(z0, min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c8(F(-(s(x1), min(s(x0), s(x1))), s(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)))) -> c8(F(-(s(s(z1)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c8(F(-(s(z1), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(x0)), s(s(z0))) -> c8(F(-(z0, min(x0, z0)), s(twice(min(s(x0), s(z0))))), TWICE(min(s(x0), s(z0))), MIN(s(x0), s(z0)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c8(F(-(s(s(z1)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c8(F(-(0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c8(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c8(F(-(x0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c8(F(-(z0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(0), s(z0)) -> c8(F(z0, s(twice(min(0, z0)))), TWICE(min(0, z0)))
   F(s(s(x0)), s(s(x1))) -> c9(F(-(s(x0), min(s(x0), s(x1))), s(s(s(twice(min(x0, x1)))))), -'(s(x0), min(s(x0), s(x1))), MIN(s(x0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c9(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), -'(s(z0), min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c9(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), -'(s(z0), min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c9(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), -'(s(s(z0)), min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c9(F(-(x0, min(x0, 0)), s(0)), -'(x0, min(x0, 0)))
   F(s(z0), s(0)) -> c9(F(-(z0, 0), s(twice(0))), -'(z0, min(z0, 0)))
   F(s(0), s(0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(0), s(x0)) -> c9(F(-(0, min(0, x0)), s(0)))
   F(s(0), s(z0)) -> c9(F(-(0, 0), s(twice(0))))
   F(s(z0), s(0)) -> c9(F(z0, s(twice(min(z0, 0)))), -'(z0, min(z0, 0)))
   F(s(s(x0)), s(s(x1))) -> c10(F(-(s(x0), min(s(x0), s(x1))), s(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)))) -> c10(F(-(s(s(z0)), min(s(s(z0)), s(s(z1)))), s(twice(s(s(min(z0, z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(s(z0)), s(s(z1))) -> c10(F(-(s(z0), s(min(z0, z1))), s(twice(s(min(z0, z1))))), TWICE(min(s(z0), s(z1))), MIN(s(z0), s(z1)))
   F(s(s(z0)), s(s(x1))) -> c10(F(-(z0, min(z0, x1)), s(twice(min(s(z0), s(x1))))), TWICE(min(s(z0), s(x1))), MIN(s(z0), s(x1)))
   F(s(s(s(z0))), s(s(s(z1)))) -> c10(F(-(s(s(z0)), s(s(min(z0, z1)))), s(twice(min(s(s(z0)), s(s(z1)))))), TWICE(min(s(s(z0)), s(s(z1)))), MIN(s(s(z0)), s(s(z1))))
   F(s(x0), s(0)) -> c10(F(-(x0, min(x0, 0)), s(0)), TWICE(min(x0, 0)))
   F(s(z0), s(0)) -> c10(F(-(z0, 0), s(twice(0))), TWICE(min(z0, 0)))
   F(s(0), s(0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, 0)))
   F(s(0), s(x0)) -> c10(F(-(0, min(0, x0)), s(0)), TWICE(min(0, x0)))
   F(s(0), s(z0)) -> c10(F(-(0, 0), s(twice(0))), TWICE(min(0, z0)))
   F(s(z0), s(0)) -> c10(F(z0, s(twice(min(z0, 0)))), TWICE(min(z0, 0)))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   MIN(s(s(y0)), s(s(y1))) -> c4(MIN(s(y0), s(y1)))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(0)), s(s(z0))) -> c7(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), -'(s(z0), min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c8(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c8(F(-(s(z0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c9(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), -'(s(z0), min(s(z0), s(0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), min(s(z0), s(0))), s(twice(s(0)))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c10(F(-(s(z0), s(0)), s(twice(min(s(z0), s(0))))), TWICE(min(s(z0), s(0))))
   F(s(s(0)), s(s(z0))) -> c10(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))), TWICE(min(s(0), s(z0))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), s(0)), s(twice(min(s(z0), s(0))))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), min(s(0), s(z0))), s(twice(s(0)))))
   F(s(s(0)), s(s(z0))) -> c9(F(-(s(0), s(0)), s(twice(min(s(0), s(z0))))))
   F(s(s(z0)), s(s(0))) -> c7(F(-(s(0), min(s(z0), s(0))), s(s(s(twice(0))))))
   TWICE(s(s(y0))) -> c6(TWICE(s(y0)))
   F(s(0), s(z0)) -> c7(F(-(z0, 0), s(0)), -'(z0, min(0, z0)))
K tuples:
   F(s(x0), s(0)) -> c8(TWICE(min(x0, 0)))
   F(s(0), s(x0)) -> c10(TWICE(min(0, x0)))
Defined Rule Symbols:   -_2, min_2, twice_1

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

Compound Symbols:   c7_3, c7_1, c7_2, c8_3, c8_2, c8_1, c9_3, c9_2, c9_1, c10_3, c10_2, c10_1, c1_1, c4_1, c6_1


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

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

(234)
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:

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

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

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

- => minus

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

(236)
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:

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

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

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

(238)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

(239) 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
   min_2
   twice_1

Due to the following rules being added:

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

And the following fresh constants:    const

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

(240)
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:

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

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

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

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

(242)
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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   min(x, 0) -> 0 [1]
   min(0, y) -> 0 [1]
   min(s(x), s(y)) -> s(min(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   f(s(x), s(0)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(0), s(y)) -> f(minus(y, 0), s(twice(0))) [3]
   f(s(s(x')), s(s(y'))) -> f(minus(s(y'), s(min(x', y'))), s(twice(s(min(x', y'))))) [3]
   f(s(x), s(0)) -> f(minus(x, 0), s(twice(0))) [3]
   f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(0), s(0)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(0), s(y)) -> f(minus(0, 0), s(twice(0))) [3]
   f(s(s(x'')), s(s(y''))) -> f(minus(s(x''), s(min(x'', y''))), s(twice(s(min(x'', y''))))) [3]
   minus(v0, v1) -> 0 [0]

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

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

(243) 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

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

(244)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(x, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 3 }-> f(minus(y, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: x >= 0, z' = 1 + 0, z = 1 + x
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z' = 1 + y, z = 1 + 0, y >= 0
   f(z, z') -{ 3 }-> f(minus(1 + x'', 1 + min(x'', y'')), 1 + twice(1 + min(x'', y''))) :|: z' = 1 + (1 + y''), y'' >= 0, x'' >= 0, z = 1 + (1 + x'')
   f(z, z') -{ 3 }-> f(minus(1 + y', 1 + min(x', y')), 1 + twice(1 + min(x', y'))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (1 + x')
   min(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   min(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y
   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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


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

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

(246)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0


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

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

   { twice }
   { minus }
   { min }
   { f }

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

(248)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

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

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

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

(250)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

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

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

(251) 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

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

(252)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

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

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

(253) 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

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

(254)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(0, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(z - 1, 0), 1 + twice(0)) :|: z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 3 }-> f(minus(z' - 1, 0), 1 + twice(0)) :|: z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(z - 1)) :|: z - 1 >= 0

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

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

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

(256)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   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}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(257) 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

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

(258)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   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}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: ?, size: O(n^1) [z]

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

(259) 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'

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

(260)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s') :|: s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s'') :|: s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(0, 0), 1 + s3) :|: s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 4 }-> f(minus(z - 1, 0), 1 + s2) :|: s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 4 }-> f(minus(z' - 1, 0), 1 + s1) :|: s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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
   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}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

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

(262)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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
   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}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

(263) 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

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

(264)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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
   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}, {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: ?, size: O(n^1) [z]

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

(265) 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'

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

(266)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> f(minus(1 + (z' - 2), 1 + min(z - 2, z' - 2)), 1 + twice(1 + min(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 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' }-> s4 :|: s4 >= 0, s4 <= 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
   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]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

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

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

(268)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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
   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]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]

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

(269) 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

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

(270)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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
   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]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  f: runtime: ?, size: O(1) [0]

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

(271) 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: ?

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

(272)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z, z') -{ 7 + s11 + s13 + 2*z' }-> f(s12, 1 + s14) :|: s11 >= 0, s11 <= z - 2, s12 >= 0, s12 <= 1 + (z' - 2), s13 >= 0, s13 <= z - 2, s14 >= 0, s14 <= 2 * (1 + s13), z - 2 >= 0, z' - 2 >= 0
   f(z, z') -{ 7 + s15 + s17 + 2*z' }-> f(s16, 1 + s18) :|: s15 >= 0, s15 <= z - 2, s16 >= 0, s16 <= 1 + (z - 2), s17 >= 0, s17 <= z - 2, s18 >= 0, s18 <= 2 * (1 + s17), z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 5 }-> f(s5, 1 + s') :|: s5 >= 0, s5 <= 0, s' >= 0, s' <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s6, 1 + s'') :|: s6 >= 0, s6 <= 0, s'' >= 0, s'' <= 2 * 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s7, 1 + s1) :|: s7 >= 0, s7 <= z' - 1, s1 >= 0, s1 <= 2 * 0, z = 1 + 0, z' - 1 >= 0
   f(z, z') -{ 5 }-> f(s8, 1 + s2) :|: s8 >= 0, s8 <= z - 1, s2 >= 0, s2 <= 2 * 0, z - 1 >= 0, z' = 1 + 0
   f(z, z') -{ 5 }-> f(s9, 1 + s3) :|: s9 >= 0, s9 <= 0, s3 >= 0, s3 <= 2 * 0, 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 + s10 :|: s10 >= 0, s10 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s4 :|: s4 >= 0, s4 <= 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
   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]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  min: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  f: runtime: INF, size: O(1) [0]

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

(273) 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]
   f(v0, v1) -> null_f [0]
   min(v0, v1) -> null_min [0]
   twice(v0) -> null_twice [0]

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

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

(274)
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:

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

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

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

(275) 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_f => 0
   null_min => 0
   null_twice => 0

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

(276)
Obligation:
Complexity RNTS consisting of the following rules:

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