KILLED
proof of input_l8QN9QDyGE.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
        (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) typed CpxTrs
    (13) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
    (14) TRS for Loop Detection
(15) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(16) CpxWeightedTrs
    (17) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (18) CpxTypedWeightedTrs
    (19) CompletionProof [UPPER BOUND(ID), 0 ms]
    (20) CpxTypedWeightedCompleteTrs
        (21) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxTypedWeightedCompleteTrs
        (23) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (26) CpxRNTS
        (27) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 377 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 122 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 447 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 1305 ms]
        (40) CpxRNTS
    (41) CompletionProof [UPPER BOUND(ID), 0 ms]
    (42) CpxTypedWeightedCompleteTrs
        (43) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
(45) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(46) CdtProblem
    (47) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
        (53) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (54) CpxTRS
        (55) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (56) CpxWeightedTrs
            (57) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (58) CpxTypedWeightedTrs
            (59) CompletionProof [UPPER BOUND(ID), 0 ms]
            (60) CpxTypedWeightedCompleteTrs
                (61) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (62) CpxTypedWeightedCompleteTrs
                (63) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (64) CpxRNTS
                (65) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (66) CpxRNTS
                (67) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (68) CpxRNTS
                (69) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (70) CpxRNTS
                (71) IntTrsBoundProof [UPPER BOUND(ID), 353 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
                (74) CpxRNTS
                (75) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (76) CpxRNTS
                (77) IntTrsBoundProof [UPPER BOUND(ID), 231 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), 777 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 2496 ms]
                (86) CpxRNTS
                (87) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (88) CpxRNTS
                (89) IntTrsBoundProof [UPPER BOUND(ID), 9611 ms]
                (90) CpxRNTS
                (91) IntTrsBoundProof [UPPER BOUND(ID), 2392 ms]
                (92) CpxRNTS
            (93) CompletionProof [UPPER BOUND(ID), 0 ms]
            (94) CpxTypedWeightedCompleteTrs
                (95) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (96) CpxRNTS
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 177 ms]
    (98) CdtProblem
        (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 47 ms]
        (100) CdtProblem
        (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms]
        (106) CdtProblem
        (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (110) CdtProblem
        (111) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
        (112) CdtProblem
        (113) CdtNarrowingProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
        (120) CdtProblem
        (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 88 ms]
        (126) CdtProblem
        (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (130) CdtProblem
        (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms]
        (140) CdtProblem
        (141) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (142) CdtProblem
        (143) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 107 ms]
        (144) CdtProblem
        (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (148) CdtProblem
        (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
        (155) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (156) CdtProblem
        (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (158) CdtProblem
        (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (160) CdtProblem
        (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
        (162) CdtProblem
        (163) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (164) CdtProblem
        (165) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 158 ms]
        (166) CdtProblem
        (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CdtProblem
        (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CdtProblem
        (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (172) CdtProblem
        (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (174) CdtProblem
        (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (176) CdtProblem
        (177) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (178) CdtProblem
        (179) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (180) CdtProblem
        (181) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (182) CdtProblem
        (183) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (184) CdtProblem
        (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (186) CdtProblem
        (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms]
        (188) CdtProblem
        (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (190) CdtProblem
        (191) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (192) CdtProblem
        (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (194) CdtProblem
        (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (196) CdtProblem
        (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (198) CdtProblem
        (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (200) CdtProblem
        (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (202) CdtProblem
        (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (204) CdtProblem
        (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (206) CdtProblem
        (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 4 ms]
        (208) CdtProblem
        (209) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (210) CdtProblem
        (211) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (212) CdtProblem
        (213) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (214) CdtProblem
        (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (216) CdtProblem
        (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (218) CdtProblem
        (219) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (220) CdtProblem


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

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


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

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:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0', S(x)) -> h(0', x)
   h(0', 0') -> 0'
   g(S(x), 0') -> 0'
   f(S(x), 0') -> 0'
   h(S(x), x2) -> h(x, x2)
   g(0', x2) -> 0'
   f(0', x2) -> 0'

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

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

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


The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x))
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x')))))
   h(0, S(x)) -> h(0, x)
   h(0, 0) -> 0
   g(S(x), 0) -> 0
   f(S(x), 0) -> 0
   h(S(x), x2) -> h(x, x2)
   g(0, x2) -> 0
   f(0, x2) -> 0

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

(5) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(6)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
   f(0, z0) -> 0
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0) -> 0
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, g_2, h_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c2, c3, c4_2, c5_2, c6, c7, c8_1, c9, c10_1


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

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

(8)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))

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

   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
   f(0, z0) -> 0
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0) -> 0
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)

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

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

(10)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0') -> c2
   F(0', z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0') -> c6
   G(0', z0) -> c7
   H(0', S(z0)) -> c8(H(0', z0))
   H(0', 0') -> c9
   H(S(z0), z1) -> c10(H(z0, z1))

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

   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0') -> 0'
   f(0', z0) -> 0'
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0') -> 0'
   g(0', z0) -> 0'
   h(0', S(z0)) -> h(0', z0)
   h(0', 0') -> 0'
   h(S(z0), z1) -> h(z0, z1)

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

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

(12)
Obligation:
Innermost TRS:
Rules:
F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
F(S(z0), 0') -> c2
F(0', z0) -> c3
G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
G(S(z0), 0') -> c6
G(0', z0) -> c7
H(0', S(z0)) -> c8(H(0', z0))
H(0', 0') -> c9
H(S(z0), z1) -> c10(H(z0, z1))
f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
f(S(z0), 0') -> 0'
f(0', z0) -> 0'
g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
g(S(z0), 0') -> 0'
g(0', z0) -> 0'
h(0', S(z0)) -> h(0', z0)
h(0', 0') -> 0'
h(S(z0), z1) -> h(z0, z1)

Types:
F :: S:0' -> S:0' -> c:c1:c2:c3
S :: S:0' -> S:0'
c :: c8:c9:c10 -> c4:c5:c6:c7 -> c:c1:c2:c3
H :: S:0' -> S:0' -> c8:c9:c10
g :: S:0' -> S:0' -> S:0'
f :: S:0' -> S:0' -> S:0'
G :: S:0' -> S:0' -> c4:c5:c6:c7
c1 :: c8:c9:c10 -> c:c1:c2:c3 -> c:c1:c2:c3
0' :: S:0'
c2 :: c:c1:c2:c3
c3 :: c:c1:c2:c3
c4 :: c8:c9:c10 -> c:c1:c2:c3 -> c4:c5:c6:c7
c5 :: c8:c9:c10 -> c4:c5:c6:c7 -> c4:c5:c6:c7
c6 :: c4:c5:c6:c7
c7 :: c4:c5:c6:c7
c8 :: c8:c9:c10 -> c8:c9:c10
c9 :: c8:c9:c10
c10 :: c8:c9:c10 -> c8:c9:c10
h :: S:0' -> S:0' -> S:0'
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_S:0'2_11 :: S:0'
hole_c8:c9:c103_11 :: c8:c9:c10
hole_c4:c5:c6:c74_11 :: c4:c5:c6:c7
gen_c:c1:c2:c35_11 :: Nat -> c:c1:c2:c3
gen_S:0'6_11 :: Nat -> S:0'
gen_c8:c9:c107_11 :: Nat -> c8:c9:c10
gen_c4:c5:c6:c78_11 :: Nat -> c4:c5:c6:c7

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

(13) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
----------------------------------------

(14)
Obligation:
Analyzing the following TRS for decreasing loops:

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


The TRS R consists of the following rules:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))

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

   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
   f(0, z0) -> 0
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0) -> 0
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)

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

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

(16)
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:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1]
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1]
   h(0, S(x)) -> h(0, x) [1]
   h(0, 0) -> 0 [1]
   g(S(x), 0) -> 0 [1]
   f(S(x), 0) -> 0 [1]
   h(S(x), x2) -> h(x, x2) [1]
   g(0, x2) -> 0 [1]
   f(0, x2) -> 0 [1]

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

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

(18)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1]
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1]
   h(0, S(x)) -> h(0, x) [1]
   h(0, 0) -> 0 [1]
   g(S(x), 0) -> 0 [1]
   f(S(x), 0) -> 0 [1]
   h(S(x), x2) -> h(x, x2) [1]
   g(0, x2) -> 0 [1]
   f(0, x2) -> 0 [1]

The TRS has the following type information:
f :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
h :: S:0 -> S:0 -> S:0
g :: S:0 -> S:0 -> S:0
0 :: S:0

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

(19) 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:
none

(c) The following functions are completely defined:

   g_2
   f_2
   h_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(20)
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:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1]
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1]
   h(0, S(x)) -> h(0, x) [1]
   h(0, 0) -> 0 [1]
   g(S(x), 0) -> 0 [1]
   f(S(x), 0) -> 0 [1]
   h(S(x), x2) -> h(x, x2) [1]
   g(0, x2) -> 0 [1]
   f(0, x2) -> 0 [1]

The TRS has the following type information:
f :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
h :: S:0 -> S:0 -> S:0
g :: S:0 -> S:0 -> S:0
0 :: S:0

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

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

(22)
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:

   f(S(S(x'')), S(S(x1))) -> h(h(f(S(x''), S(S(x1))), g(x'', S(S(S(S(x1)))))), h(g(S(S(S(x''))), S(x1)), f(S(S(S(S(S(S(x'')))))), x1))) [3]
   f(S(S(x'')), S(0)) -> h(h(f(S(x''), S(0)), g(x'', S(S(S(0))))), 0) [3]
   f(S(0), S(S(x3))) -> h(0, h(g(S(S(0)), S(x3)), f(S(S(S(S(S(0))))), x3))) [3]
   f(S(0), S(0)) -> h(0, 0) [3]
   g(S(S(x4)), S(x')) -> h(h(g(S(x4), S(x')), f(S(S(S(S(x4)))), x')), h(f(S(x4), S(S(S(x')))), g(x4, S(S(S(S(S(x')))))))) [3]
   g(S(0), S(x')) -> h(h(g(0, S(x')), f(S(S(S(0))), x')), 0) [3]
   h(0, S(x)) -> h(0, x) [1]
   h(0, 0) -> 0 [1]
   g(S(x), 0) -> 0 [1]
   f(S(x), 0) -> 0 [1]
   h(S(x), x2) -> h(x, x2) [1]
   g(0, x2) -> 0 [1]
   f(0, x2) -> 0 [1]

The TRS has the following type information:
f :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
h :: S:0 -> S:0 -> S:0
g :: S:0 -> S:0 -> S:0
0 :: S:0

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

(23) 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

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + x'', 1 + 0), g(x'', 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, x'' >= 0, z = 1 + (1 + x'')
   f(z, z') -{ 3 }-> h(h(f(1 + x'', 1 + (1 + x1)), g(x'', 1 + (1 + (1 + (1 + x1))))), h(g(1 + (1 + (1 + x'')), 1 + x1), f(1 + (1 + (1 + (1 + (1 + (1 + x''))))), x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x'' >= 0, z = 1 + (1 + x'')
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + x3), f(1 + (1 + (1 + (1 + (1 + 0)))), x3))) :|: z' = 1 + (1 + x3), z = 1 + 0, x3 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + x'), f(1 + (1 + (1 + 0)), x')), 0) :|: z' = 1 + x', z = 1 + 0, x' >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + x4, 1 + x'), f(1 + (1 + (1 + (1 + x4))), x')), h(f(1 + x4, 1 + (1 + (1 + x'))), g(x4, 1 + (1 + (1 + (1 + (1 + x'))))))) :|: x4 >= 0, z = 1 + (1 + x4), z' = 1 + x', x' >= 0
   g(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0
   h(z, z') -{ 1 }-> h(x, x2) :|: z' = x2, x >= 0, z = 1 + x, x2 >= 0
   h(z, z') -{ 1 }-> h(0, x) :|: z' = 1 + x, x >= 0, z = 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0


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

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0


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

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

   { h }
   { f, g }

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {h}, {f,g}

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

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {h}, {f,g}

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {h}, {f,g}
Previous analysis results are:
  h: runtime: ?, size: O(1) [0]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 3 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 1 }-> h(z - 1, z') :|: z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:
  h: runtime: O(n^1) [1 + z + z'], size: O(1) [0]

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

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

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

   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:
  h: runtime: O(n^1) [1 + z + z'], size: O(1) [0]

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

(37) 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

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

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

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

   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:
  h: runtime: O(n^1) [1 + z + z'], size: O(1) [0]
  f: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(39) 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: ?

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

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

   f(z, z') -{ 4 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 3 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   g(z, z') -{ 3 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z = 1 + 0, z' - 1 >= 0
   g(z, z') -{ 3 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z - 2 >= 0, z' - 1 >= 0
   g(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z = 0, z' >= 0
   h(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 1 + z + z' }-> s'' :|: s'' >= 0, s'' <= 0, z - 1 >= 0, z' >= 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Function symbols to be analyzed: {f,g}
Previous analysis results are:
  h: runtime: O(n^1) [1 + z + z'], size: O(1) [0]
  f: runtime: INF, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(41) 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:
none

And the following fresh constants: none

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

(42)
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:

   f(S(x'), S(x)) -> h(g(x', S(x)), f(S(S(S(x'))), x)) [1]
   g(S(x), S(x')) -> h(f(S(x), S(x')), g(x, S(S(S(x'))))) [1]
   h(0, S(x)) -> h(0, x) [1]
   h(0, 0) -> 0 [1]
   g(S(x), 0) -> 0 [1]
   f(S(x), 0) -> 0 [1]
   h(S(x), x2) -> h(x, x2) [1]
   g(0, x2) -> 0 [1]
   f(0, x2) -> 0 [1]

The TRS has the following type information:
f :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
h :: S:0 -> S:0 -> S:0
g :: S:0 -> S:0 -> S:0
0 :: S:0

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

(43) 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

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

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

   f(z, z') -{ 1 }-> h(g(x', 1 + x), f(1 + (1 + (1 + x')), x)) :|: z = 1 + x', z' = 1 + x, x' >= 0, x >= 0
   f(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   f(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0
   g(z, z') -{ 1 }-> h(f(1 + x, 1 + x'), g(x, 1 + (1 + (1 + x')))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x
   g(z, z') -{ 1 }-> 0 :|: x >= 0, z = 1 + x, z' = 0
   g(z, z') -{ 1 }-> 0 :|: z' = x2, z = 0, x2 >= 0
   h(z, z') -{ 1 }-> h(x, x2) :|: z' = x2, x >= 0, z = 1 + x, x2 >= 0
   h(z, z') -{ 1 }-> h(0, x) :|: z' = 1 + x, x >= 0, z = 0
   h(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
   f(0, z0) -> 0
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0) -> 0
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   F(S(z0), 0) -> c2
   F(0, z0) -> c3
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(z0), 0) -> c6
   G(0, z0) -> c7
   H(0, S(z0)) -> c8(H(0, z0))
   H(0, 0) -> c9
   H(S(z0), z1) -> c10(H(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, g_2, h_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c2, c3, c4_2, c5_2, c6, c7, c8_1, c9, c10_1


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

(47) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   G(0, z0) -> c7
   H(0, 0) -> c9
   F(0, z0) -> c3
   F(S(z0), 0) -> c2
   G(S(z0), 0) -> c6

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
   f(0, z0) -> 0
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(S(z0), 0) -> 0
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
K tuples:none
Defined Rule Symbols:   f_2, g_2, h_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c4_2, c5_2, c8_1, c10_1


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

(49) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0, z0) -> 0
   g(S(z0), 0) -> 0

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c4_2, c5_2, c8_1, c10_1


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

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

(52)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))

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

   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0

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

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

(54)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0

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

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

(56)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) [1]
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) [1]
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) [1]
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(0, z0) -> 0 [0]
   h(0, S(z0)) -> h(0, z0) [0]
   h(0, 0) -> 0 [0]
   h(S(z0), z1) -> h(z0, z1) [0]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]

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

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

(58)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) [1]
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) [1]
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) [1]
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(0, z0) -> 0 [0]
   h(0, S(z0)) -> h(0, z0) [0]
   h(0, 0) -> 0 [0]
   h(S(z0), z1) -> h(z0, z1) [0]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]

The TRS has the following type information:
F :: S:0 -> S:0 -> c:c1
S :: S:0 -> S:0
c :: c8:c10 -> c4:c5 -> c:c1
H :: S:0 -> S:0 -> c8:c10
g :: S:0 -> S:0 -> S:0
f :: S:0 -> S:0 -> S:0
G :: S:0 -> S:0 -> c4:c5
c1 :: c8:c10 -> c:c1 -> c:c1
c4 :: c8:c10 -> c:c1 -> c4:c5
c5 :: c8:c10 -> c4:c5 -> c4:c5
0 :: S:0
c8 :: c8:c10 -> c8:c10
c10 :: c8:c10 -> c8:c10
h :: S:0 -> S:0 -> S:0

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

(59) 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
   G_2
   H_2

(c) The following functions are completely defined:

   g_2
   h_2
   f_2

Due to the following rules being added:

   g(v0, v1) -> 0 [0]
   h(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

And the following fresh constants:    const, const1, const2

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

(60)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) [1]
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) [1]
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) [1]
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(0, z0) -> 0 [0]
   h(0, S(z0)) -> h(0, z0) [0]
   h(0, 0) -> 0 [0]
   h(S(z0), z1) -> h(z0, z1) [0]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   g(v0, v1) -> 0 [0]
   h(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

The TRS has the following type information:
F :: S:0 -> S:0 -> c:c1
S :: S:0 -> S:0
c :: c8:c10 -> c4:c5 -> c:c1
H :: S:0 -> S:0 -> c8:c10
g :: S:0 -> S:0 -> S:0
f :: S:0 -> S:0 -> S:0
G :: S:0 -> S:0 -> c4:c5
c1 :: c8:c10 -> c:c1 -> c:c1
c4 :: c8:c10 -> c:c1 -> c4:c5
c5 :: c8:c10 -> c4:c5 -> c4:c5
0 :: S:0
c8 :: c8:c10 -> c8:c10
c10 :: c8:c10 -> c8:c10
h :: S:0 -> S:0 -> S:0
const :: c:c1
const1 :: c8:c10
const2 :: c4:c5

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

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

(62)
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:

   F(S(S(z0')), S(S(z1'))) -> c(H(h(f(S(z0'), S(S(z1'))), g(z0', S(S(S(S(z1')))))), h(g(S(S(S(z0'))), S(z1')), f(S(S(S(S(S(S(z0')))))), z1'))), G(S(z0'), S(S(z1')))) [1]
   F(S(S(z0')), S(0)) -> c(H(h(f(S(z0'), S(0)), g(z0', S(S(S(0))))), 0), G(S(z0'), S(0))) [1]
   F(S(S(z0')), S(z1)) -> c(H(h(f(S(z0'), S(z1)), g(z0', S(S(S(z1))))), 0), G(S(z0'), S(z1))) [1]
   F(S(0), S(S(z1''))) -> c(H(0, h(g(S(S(0)), S(z1'')), f(S(S(S(S(S(0))))), z1''))), G(0, S(S(z1'')))) [1]
   F(S(0), S(0)) -> c(H(0, 0), G(0, S(0))) [1]
   F(S(0), S(z1)) -> c(H(0, 0), G(0, S(z1))) [1]
   F(S(z0), S(S(z11))) -> c(H(0, h(g(S(S(z0)), S(z11)), f(S(S(S(S(S(z0))))), z11))), G(z0, S(S(z11)))) [1]
   F(S(z0), S(0)) -> c(H(0, 0), G(z0, S(0))) [1]
   F(S(z0), S(z1)) -> c(H(0, 0), G(z0, S(z1))) [1]
   F(S(S(z0'')), S(S(z12))) -> c1(H(h(f(S(z0''), S(S(z12))), g(z0'', S(S(S(S(z12)))))), h(g(S(S(S(z0''))), S(z12)), f(S(S(S(S(S(S(z0'')))))), z12))), F(S(S(S(S(z0'')))), S(z12))) [1]
   F(S(S(z0'')), S(0)) -> c1(H(h(f(S(z0''), S(0)), g(z0'', S(S(S(0))))), 0), F(S(S(S(S(z0'')))), 0)) [1]
   F(S(S(z0'')), S(z1)) -> c1(H(h(f(S(z0''), S(z1)), g(z0'', S(S(S(z1))))), 0), F(S(S(S(S(z0'')))), z1)) [1]
   F(S(0), S(S(z13))) -> c1(H(0, h(g(S(S(0)), S(z13)), f(S(S(S(S(S(0))))), z13))), F(S(S(S(0))), S(z13))) [1]
   F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0)) [1]
   F(S(0), S(z1)) -> c1(H(0, 0), F(S(S(S(0))), z1)) [1]
   F(S(z0), S(S(z14))) -> c1(H(0, h(g(S(S(z0)), S(z14)), f(S(S(S(S(S(z0))))), z14))), F(S(S(S(z0))), S(z14))) [1]
   F(S(z0), S(0)) -> c1(H(0, 0), F(S(S(S(z0))), 0)) [1]
   F(S(z0), S(z1)) -> c1(H(0, 0), F(S(S(S(z0))), z1)) [1]
   G(S(S(z01)), S(z1)) -> c4(H(h(g(S(z01), S(z1)), f(S(S(S(S(z01)))), z1)), h(f(S(z01), S(S(S(z1)))), g(z01, S(S(S(S(S(z1)))))))), F(S(S(z01)), S(z1))) [1]
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1))) [1]
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), 0), F(S(z0), S(z1))) [1]
   G(S(S(z02)), S(z1)) -> c4(H(0, h(f(S(z02), S(S(S(z1)))), g(z02, S(S(S(S(S(z1)))))))), F(S(S(z02)), S(z1))) [1]
   G(S(0), S(z1)) -> c4(H(0, 0), F(S(0), S(z1))) [1]
   G(S(z0), S(z1)) -> c4(H(0, 0), F(S(z0), S(z1))) [1]
   G(S(S(z03)), S(z1)) -> c5(H(h(g(S(z03), S(z1)), f(S(S(S(S(z03)))), z1)), h(f(S(z03), S(S(S(z1)))), g(z03, S(S(S(S(S(z1)))))))), G(S(z03), S(S(S(z1))))) [1]
   G(S(0), S(z1)) -> c5(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), G(0, S(S(S(z1))))) [1]
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), 0), G(z0, S(S(S(z1))))) [1]
   G(S(S(z04)), S(z1)) -> c5(H(0, h(f(S(z04), S(S(S(z1)))), g(z04, S(S(S(S(S(z1)))))))), G(S(z04), S(S(S(z1))))) [1]
   G(S(0), S(z1)) -> c5(H(0, 0), G(0, S(S(S(z1))))) [1]
   G(S(z0), S(z1)) -> c5(H(0, 0), G(z0, S(S(S(z1))))) [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   g(S(S(z05)), S(z1)) -> h(h(g(S(z05), S(z1)), f(S(S(S(S(z05)))), z1)), h(f(S(z05), S(S(S(z1)))), g(z05, S(S(S(S(S(z1)))))))) [0]
   g(S(0), S(z1)) -> h(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0) [0]
   g(S(z0), S(z1)) -> h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), 0) [0]
   g(S(S(z06)), S(z1)) -> h(0, h(f(S(z06), S(S(S(z1)))), g(z06, S(S(S(S(S(z1)))))))) [0]
   g(S(0), S(z1)) -> h(0, 0) [0]
   g(S(z0), S(z1)) -> h(0, 0) [0]
   g(0, z0) -> 0 [0]
   h(0, S(z0)) -> h(0, z0) [0]
   h(0, 0) -> 0 [0]
   h(S(z0), z1) -> h(z0, z1) [0]
   f(S(S(z07)), S(S(z15))) -> h(h(f(S(z07), S(S(z15))), g(z07, S(S(S(S(z15)))))), h(g(S(S(S(z07))), S(z15)), f(S(S(S(S(S(S(z07)))))), z15))) [0]
   f(S(S(z07)), S(0)) -> h(h(f(S(z07), S(0)), g(z07, S(S(S(0))))), 0) [0]
   f(S(S(z07)), S(z1)) -> h(h(f(S(z07), S(z1)), g(z07, S(S(S(z1))))), 0) [0]
   f(S(0), S(S(z16))) -> h(0, h(g(S(S(0)), S(z16)), f(S(S(S(S(S(0))))), z16))) [0]
   f(S(0), S(0)) -> h(0, 0) [0]
   f(S(0), S(z1)) -> h(0, 0) [0]
   f(S(z0), S(S(z17))) -> h(0, h(g(S(S(z0)), S(z17)), f(S(S(S(S(S(z0))))), z17))) [0]
   f(S(z0), S(0)) -> h(0, 0) [0]
   f(S(z0), S(z1)) -> h(0, 0) [0]
   f(S(z0), 0) -> 0 [0]
   g(v0, v1) -> 0 [0]
   h(v0, v1) -> 0 [0]
   f(v0, v1) -> 0 [0]

The TRS has the following type information:
F :: S:0 -> S:0 -> c:c1
S :: S:0 -> S:0
c :: c8:c10 -> c4:c5 -> c:c1
H :: S:0 -> S:0 -> c8:c10
g :: S:0 -> S:0 -> S:0
f :: S:0 -> S:0 -> S:0
G :: S:0 -> S:0 -> c4:c5
c1 :: c8:c10 -> c:c1 -> c:c1
c4 :: c8:c10 -> c:c1 -> c4:c5
c5 :: c8:c10 -> c4:c5 -> c4:c5
0 :: S:0
c8 :: c8:c10 -> c8:c10
c10 :: c8:c10 -> c8:c10
h :: S:0 -> S:0 -> S:0
const :: c:c1
const1 :: c8:c10
const2 :: c4:c5

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

(63) 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

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0', 1 + z1), g(z0', 1 + (1 + (1 + z1)))), 0) + G(1 + z0', 1 + z1) :|: z1 >= 0, z = 1 + (1 + z0'), z0' >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0', 1 + 0), g(z0', 1 + (1 + (1 + 0)))), 0) + G(1 + z0', 1 + 0) :|: z = 1 + (1 + z0'), z0' >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0', 1 + (1 + z1')), g(z0', 1 + (1 + (1 + (1 + z1'))))), h(g(1 + (1 + (1 + z0')), 1 + z1'), f(1 + (1 + (1 + (1 + (1 + (1 + z0'))))), z1'))) + G(1 + z0', 1 + (1 + z1')) :|: z' = 1 + (1 + z1'), z = 1 + (1 + z0'), z0' >= 0, z1' >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0'', 1 + z1), g(z0'', 1 + (1 + (1 + z1)))), 0) + F(1 + (1 + (1 + (1 + z0''))), z1) :|: z1 >= 0, z = 1 + (1 + z0''), z0'' >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0'', 1 + 0), g(z0'', 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + z0''))), 0) :|: z = 1 + (1 + z0''), z' = 1 + 0, z0'' >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + z0'', 1 + (1 + z12)), g(z0'', 1 + (1 + (1 + (1 + z12))))), h(g(1 + (1 + (1 + z0'')), 1 + z12), f(1 + (1 + (1 + (1 + (1 + (1 + z0''))))), z12))) + F(1 + (1 + (1 + (1 + z0''))), 1 + z12) :|: z' = 1 + (1 + z12), z = 1 + (1 + z0''), z12 >= 0, z0'' >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + z0), 1 + z11), f(1 + (1 + (1 + (1 + (1 + z0)))), z11))) + G(z0, 1 + (1 + z11)) :|: z11 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + (1 + z11)
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + z0), 1 + z14), f(1 + (1 + (1 + (1 + (1 + z0)))), z14))) + F(1 + (1 + (1 + z0)), 1 + z14) :|: z' = 1 + (1 + z14), z = 1 + z0, z0 >= 0, z14 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + z1''), f(1 + (1 + (1 + (1 + (1 + 0)))), z1''))) + G(0, 1 + (1 + z1'')) :|: z' = 1 + (1 + z1''), z = 1 + 0, z1'' >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + z13), f(1 + (1 + (1 + (1 + (1 + 0)))), z13))) + F(1 + (1 + (1 + 0)), 1 + z13) :|: z' = 1 + (1 + z13), z = 1 + 0, z13 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z0, 1 + 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + z0)), z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + z0)), 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z1) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)), 0) + G(z0, 1 + (1 + (1 + z1))) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(h(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)), 0) + F(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + z1), f(1 + (1 + (1 + 0)), z1)), 0) + G(0, 1 + (1 + (1 + z1))) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + z1), f(1 + (1 + (1 + 0)), z1)), 0) + F(1 + 0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + z01, 1 + z1), f(1 + (1 + (1 + (1 + z01))), z1)), h(f(1 + z01, 1 + (1 + (1 + z1))), g(z01, 1 + (1 + (1 + (1 + (1 + z1))))))) + F(1 + (1 + z01), 1 + z1) :|: z = 1 + (1 + z01), z1 >= 0, z01 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + z03, 1 + z1), f(1 + (1 + (1 + (1 + z03))), z1)), h(f(1 + z03, 1 + (1 + (1 + z1))), g(z03, 1 + (1 + (1 + (1 + (1 + z1))))))) + G(1 + z03, 1 + (1 + (1 + z1))) :|: z = 1 + (1 + z03), z1 >= 0, z03 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + z02, 1 + (1 + (1 + z1))), g(z02, 1 + (1 + (1 + (1 + (1 + z1))))))) + F(1 + (1 + z02), 1 + z1) :|: z = 1 + (1 + z02), z1 >= 0, z02 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + z04, 1 + (1 + (1 + z1))), g(z04, 1 + (1 + (1 + (1 + (1 + z1))))))) + G(1 + z04, 1 + (1 + (1 + z1))) :|: z04 >= 0, z1 >= 0, z = 1 + (1 + z04), z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z0, 1 + (1 + (1 + z1))) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + z1))) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + z1) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   H(z, z') -{ 1 }-> 1 + H(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   f(z, z') -{ 0 }-> h(h(f(1 + z07, 1 + z1), g(z07, 1 + (1 + (1 + z1)))), 0) :|: z1 >= 0, z = 1 + (1 + z07), z07 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> h(h(f(1 + z07, 1 + 0), g(z07, 1 + (1 + (1 + 0)))), 0) :|: z = 1 + (1 + z07), z07 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + z07, 1 + (1 + z15)), g(z07, 1 + (1 + (1 + (1 + z15))))), h(g(1 + (1 + (1 + z07)), 1 + z15), f(1 + (1 + (1 + (1 + (1 + (1 + z07))))), z15))) :|: z' = 1 + (1 + z15), z15 >= 0, z = 1 + (1 + z07), z07 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + z0), 1 + z17), f(1 + (1 + (1 + (1 + (1 + z0)))), z17))) :|: z' = 1 + (1 + z17), z17 >= 0, z = 1 + z0, z0 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + z16), f(1 + (1 + (1 + (1 + (1 + 0)))), z16))) :|: z' = 1 + (1 + z16), z = 1 + 0, z16 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + z0, z' = 1 + 0, z0 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 0 }-> h(h(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)), 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   g(z, z') -{ 0 }-> h(h(g(0, 1 + z1), f(1 + (1 + (1 + 0)), z1)), 0) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   g(z, z') -{ 0 }-> h(h(g(1 + z05, 1 + z1), f(1 + (1 + (1 + (1 + z05))), z1)), h(f(1 + z05, 1 + (1 + (1 + z1))), g(z05, 1 + (1 + (1 + (1 + (1 + z1))))))) :|: z1 >= 0, z05 >= 0, z' = 1 + z1, z = 1 + (1 + z05)
   g(z, z') -{ 0 }-> h(0, h(f(1 + z06, 1 + (1 + (1 + z1))), g(z06, 1 + (1 + (1 + (1 + (1 + z1))))))) :|: z1 >= 0, z06 >= 0, z = 1 + (1 + z06), z' = 1 + z1
   g(z, z') -{ 0 }-> h(0, 0) :|: z1 >= 0, z = 1 + 0, z' = 1 + z1
   g(z, z') -{ 0 }-> h(0, 0) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   g(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   h(z, z') -{ 0 }-> h(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   h(z, z') -{ 0 }-> h(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> h(z - 1, z') :|: z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { h }
   { H }
   { f, g }
   { G, F }

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> h(z - 1, z') :|: z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {h}, {H}, {f,g}, {G,F}

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

(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:

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> h(z - 1, z') :|: z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {h}, {H}, {f,g}, {G,F}

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

(71) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: h
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:

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> h(z - 1, z') :|: z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {h}, {H}, {f,g}, {G,F}
Previous analysis results are:
  h: runtime: ?, size: O(1) [0]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(0, 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> h(0, z' - 1) :|: z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> h(z - 1, z') :|: z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {H}, {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], 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:

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {H}, {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]

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

(77) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {H}, {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: ?, size: O(1) [0]

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

(79) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), 0) :|: z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + 0)), z' - 1) :|: z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), 0) :|: z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(0, 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z' - 1) :|: z' - 1 >= 0, z = 0
   H(z, z') -{ 1 }-> 1 + H(z - 1, z') :|: z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]

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

(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:

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]

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

(83) 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

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

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

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

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {f,g}, {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]
  f: runtime: ?, size: O(1) [0]
  g: runtime: ?, size: O(1) [0]

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

(85) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed RUNTIME bound using CoFloCo for: g
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:

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + G(1 + (z - 2), 1 + 0) :|: z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + G(1 + (z - 2), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + G(0, 1 + (1 + (z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + G(z - 1, 1 + (1 + (z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + H(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: z - 1 >= 0, z' - 2 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + G(0, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) + F(1 + 0, 1 + (z' - 1)) :|: z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) + F(1 + (z - 1), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + H(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + 0), g(z - 2, 1 + (1 + (1 + 0)))), 0) :|: z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (z' - 1)), g(z - 2, 1 + (1 + (1 + (z' - 1))))), 0) :|: z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(h(f(1 + (z - 2), 1 + (1 + (z' - 2))), g(z - 2, 1 + (1 + (1 + (1 + (z' - 2)))))), h(g(1 + (1 + (1 + (z - 2))), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (1 + (z - 2)))))), z' - 2))) :|: z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + 0), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + 0)))), z' - 2))) :|: z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> h(0, h(g(1 + (1 + (z - 1)), 1 + (z' - 2)), f(1 + (1 + (1 + (1 + (1 + (z - 1))))), z' - 2))) :|: z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(0, 1 + (z' - 1)), f(1 + (1 + (1 + 0)), z' - 1)), 0) :|: z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> h(h(g(z - 1, 1 + (z' - 1)), f(1 + (1 + (1 + (z - 1))), z' - 1)), 0) :|: z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> h(h(g(1 + (z - 2), 1 + (z' - 1)), f(1 + (1 + (1 + (1 + (z - 2)))), z' - 1)), h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> h(0, h(f(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))), g(z - 2, 1 + (1 + (1 + (1 + (1 + (z' - 1)))))))) :|: z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]

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

(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:

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + s25 }-> 1 + s26 + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: s20 >= 0, s20 <= 0, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, s23 >= 0, s23 <= 0, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= 0, s26 >= 0, s26 <= 0, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 + s33 }-> 1 + s34 + G(1 + (z - 2), 1 + (z' - 1)) :|: s31 >= 0, s31 <= 0, s32 >= 0, s32 <= 0, s33 >= 0, s33 <= 0, s34 >= 0, s34 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s37 }-> 1 + s38 + G(0, 1 + (1 + (z' - 2))) :|: s35 >= 0, s35 <= 0, s36 >= 0, s36 <= 0, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s41 }-> 1 + s42 + G(z - 1, 1 + (1 + (z' - 2))) :|: s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= 0, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s45 + s48 }-> 1 + s49 + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, s45 >= 0, s45 <= 0, s46 >= 0, s46 <= 0, s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 0, s49 >= 0, s49 <= 0, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s52 }-> 1 + s53 + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: s50 >= 0, s50 <= 0, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= 0, s53 >= 0, s53 <= 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 + s56 }-> 1 + s57 + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: s54 >= 0, s54 <= 0, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, s57 >= 0, s57 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 + s60 }-> 1 + s61 + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: s58 >= 0, s58 <= 0, s59 >= 0, s59 <= 0, s60 >= 0, s60 <= 0, s61 >= 0, s61 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s64 }-> 1 + s65 + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: s62 >= 0, s62 <= 0, s63 >= 0, s63 <= 0, s64 >= 0, s64 <= 0, s65 >= 0, s65 <= 0, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s102 }-> 1 + s103 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s100 >= 0, s100 <= 0, s101 >= 0, s101 <= 0, s102 >= 0, s102 <= 0, s103 >= 0, s103 <= 0, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s68 + s71 }-> 1 + s72 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s66 >= 0, s66 <= 0, s67 >= 0, s67 <= 0, s68 >= 0, s68 <= 0, s69 >= 0, s69 <= 0, s70 >= 0, s70 <= 0, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s79 }-> 1 + s80 + F(1 + (z - 1), 1 + (z' - 1)) :|: s77 >= 0, s77 <= 0, s78 >= 0, s78 <= 0, s79 >= 0, s79 <= 0, s80 >= 0, s80 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s83 }-> 1 + s84 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s81 >= 0, s81 <= 0, s82 >= 0, s82 <= 0, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s87 + s90 }-> 1 + s91 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s85 >= 0, s85 <= 0, s86 >= 0, s86 <= 0, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, s89 >= 0, s89 <= 0, s90 >= 0, s90 <= 0, s91 >= 0, s91 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s94 }-> 1 + s95 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s92 >= 0, s92 <= 0, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, s95 >= 0, s95 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s98 }-> 1 + s99 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s96 >= 0, s96 <= 0, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= 0, s99 >= 0, s99 <= 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s129 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s133 :|: s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s110 :|: s104 >= 0, s104 <= 0, s105 >= 0, s105 <= 0, s106 >= 0, s106 <= 0, s107 >= 0, s107 <= 0, s108 >= 0, s108 <= 0, s109 >= 0, s109 <= 0, s110 >= 0, s110 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s122 :|: s119 >= 0, s119 <= 0, s120 >= 0, s120 <= 0, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]

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

(89) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + s25 }-> 1 + s26 + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: s20 >= 0, s20 <= 0, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, s23 >= 0, s23 <= 0, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= 0, s26 >= 0, s26 <= 0, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 + s33 }-> 1 + s34 + G(1 + (z - 2), 1 + (z' - 1)) :|: s31 >= 0, s31 <= 0, s32 >= 0, s32 <= 0, s33 >= 0, s33 <= 0, s34 >= 0, s34 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s37 }-> 1 + s38 + G(0, 1 + (1 + (z' - 2))) :|: s35 >= 0, s35 <= 0, s36 >= 0, s36 <= 0, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s41 }-> 1 + s42 + G(z - 1, 1 + (1 + (z' - 2))) :|: s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= 0, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s45 + s48 }-> 1 + s49 + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, s45 >= 0, s45 <= 0, s46 >= 0, s46 <= 0, s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 0, s49 >= 0, s49 <= 0, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s52 }-> 1 + s53 + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: s50 >= 0, s50 <= 0, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= 0, s53 >= 0, s53 <= 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 + s56 }-> 1 + s57 + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: s54 >= 0, s54 <= 0, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, s57 >= 0, s57 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 + s60 }-> 1 + s61 + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: s58 >= 0, s58 <= 0, s59 >= 0, s59 <= 0, s60 >= 0, s60 <= 0, s61 >= 0, s61 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s64 }-> 1 + s65 + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: s62 >= 0, s62 <= 0, s63 >= 0, s63 <= 0, s64 >= 0, s64 <= 0, s65 >= 0, s65 <= 0, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s102 }-> 1 + s103 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s100 >= 0, s100 <= 0, s101 >= 0, s101 <= 0, s102 >= 0, s102 <= 0, s103 >= 0, s103 <= 0, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s68 + s71 }-> 1 + s72 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s66 >= 0, s66 <= 0, s67 >= 0, s67 <= 0, s68 >= 0, s68 <= 0, s69 >= 0, s69 <= 0, s70 >= 0, s70 <= 0, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s79 }-> 1 + s80 + F(1 + (z - 1), 1 + (z' - 1)) :|: s77 >= 0, s77 <= 0, s78 >= 0, s78 <= 0, s79 >= 0, s79 <= 0, s80 >= 0, s80 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s83 }-> 1 + s84 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s81 >= 0, s81 <= 0, s82 >= 0, s82 <= 0, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s87 + s90 }-> 1 + s91 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s85 >= 0, s85 <= 0, s86 >= 0, s86 <= 0, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, s89 >= 0, s89 <= 0, s90 >= 0, s90 <= 0, s91 >= 0, s91 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s94 }-> 1 + s95 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s92 >= 0, s92 <= 0, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, s95 >= 0, s95 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s98 }-> 1 + s99 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s96 >= 0, s96 <= 0, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= 0, s99 >= 0, s99 <= 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s129 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s133 :|: s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s110 :|: s104 >= 0, s104 <= 0, s105 >= 0, s105 <= 0, s106 >= 0, s106 <= 0, s107 >= 0, s107 <= 0, s108 >= 0, s108 <= 0, s109 >= 0, s109 <= 0, s110 >= 0, s110 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s122 :|: s119 >= 0, s119 <= 0, s120 >= 0, s120 <= 0, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]
  G: runtime: ?, size: O(1) [0]
  F: runtime: ?, size: O(n^1) [2 + z']

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

(91) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   F(z, z') -{ 1 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s22 + s25 }-> 1 + s26 + G(1 + (z - 2), 1 + (1 + (z' - 2))) :|: s20 >= 0, s20 <= 0, s21 >= 0, s21 <= 0, s22 >= 0, s22 <= 0, s23 >= 0, s23 <= 0, s24 >= 0, s24 <= 0, s25 >= 0, s25 <= 0, s26 >= 0, s26 <= 0, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 1 + s33 }-> 1 + s34 + G(1 + (z - 2), 1 + (z' - 1)) :|: s31 >= 0, s31 <= 0, s32 >= 0, s32 <= 0, s33 >= 0, s33 <= 0, s34 >= 0, s34 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s37 }-> 1 + s38 + G(0, 1 + (1 + (z' - 2))) :|: s35 >= 0, s35 <= 0, s36 >= 0, s36 <= 0, s37 >= 0, s37 <= 0, s38 >= 0, s38 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s41 }-> 1 + s42 + G(z - 1, 1 + (1 + (z' - 2))) :|: s39 >= 0, s39 <= 0, s40 >= 0, s40 <= 0, s41 >= 0, s41 <= 0, s42 >= 0, s42 <= 0, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 1 + s45 + s48 }-> 1 + s49 + F(1 + (1 + (1 + (1 + (z - 2)))), 1 + (z' - 2)) :|: s43 >= 0, s43 <= 0, s44 >= 0, s44 <= 0, s45 >= 0, s45 <= 0, s46 >= 0, s46 <= 0, s47 >= 0, s47 <= 0, s48 >= 0, s48 <= 0, s49 >= 0, s49 <= 0, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 1 + s52 }-> 1 + s53 + F(1 + (1 + (1 + (1 + (z - 2)))), 0) :|: s50 >= 0, s50 <= 0, s51 >= 0, s51 <= 0, s52 >= 0, s52 <= 0, s53 >= 0, s53 <= 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 1 + s56 }-> 1 + s57 + F(1 + (1 + (1 + (1 + (z - 2)))), z' - 1) :|: s54 >= 0, s54 <= 0, s55 >= 0, s55 <= 0, s56 >= 0, s56 <= 0, s57 >= 0, s57 <= 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 1 + s60 }-> 1 + s61 + F(1 + (1 + (1 + 0)), 1 + (z' - 2)) :|: s58 >= 0, s58 <= 0, s59 >= 0, s59 <= 0, s60 >= 0, s60 <= 0, s61 >= 0, s61 <= 0, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 1 + s64 }-> 1 + s65 + F(1 + (1 + (1 + (z - 1))), 1 + (z' - 2)) :|: s62 >= 0, s62 <= 0, s63 >= 0, s63 <= 0, s64 >= 0, s64 <= 0, s65 >= 0, s65 <= 0, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 1 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 1 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 1 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s102 }-> 1 + s103 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s100 >= 0, s100 <= 0, s101 >= 0, s101 <= 0, s102 >= 0, s102 <= 0, s103 >= 0, s103 <= 0, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s68 + s71 }-> 1 + s72 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s66 >= 0, s66 <= 0, s67 >= 0, s67 <= 0, s68 >= 0, s68 <= 0, s69 >= 0, s69 <= 0, s70 >= 0, s70 <= 0, s71 >= 0, s71 <= 0, s72 >= 0, s72 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s79 }-> 1 + s80 + F(1 + (z - 1), 1 + (z' - 1)) :|: s77 >= 0, s77 <= 0, s78 >= 0, s78 <= 0, s79 >= 0, s79 <= 0, s80 >= 0, s80 <= 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 + s83 }-> 1 + s84 + F(1 + (1 + (z - 2)), 1 + (z' - 1)) :|: s81 >= 0, s81 <= 0, s82 >= 0, s82 <= 0, s83 >= 0, s83 <= 0, s84 >= 0, s84 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s87 + s90 }-> 1 + s91 + G(1 + (z - 2), 1 + (1 + (1 + (z' - 1)))) :|: s85 >= 0, s85 <= 0, s86 >= 0, s86 <= 0, s87 >= 0, s87 <= 0, s88 >= 0, s88 <= 0, s89 >= 0, s89 <= 0, s90 >= 0, s90 <= 0, s91 >= 0, s91 <= 0, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 1 + s94 }-> 1 + s95 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s92 >= 0, s92 <= 0, s93 >= 0, s93 <= 0, s94 >= 0, s94 <= 0, s95 >= 0, s95 <= 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 1 + s98 }-> 1 + s99 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s96 >= 0, s96 <= 0, s97 >= 0, s97 <= 0, s98 >= 0, s98 <= 0, s99 >= 0, s99 <= 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0, z' - 1 >= 0, z = 0
   H(z, z') -{ z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= 0, z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s129 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, z' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s133 :|: s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z - 2 >= 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z = 1 + 0, z' - 2 >= 0
   f(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 2 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s110 :|: s104 >= 0, s104 <= 0, s105 >= 0, s105 <= 0, s106 >= 0, s106 <= 0, s107 >= 0, s107 <= 0, s108 >= 0, s108 <= 0, s109 >= 0, s109 <= 0, s110 >= 0, s110 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s122 :|: s119 >= 0, s119 <= 0, s120 >= 0, s120 <= 0, s121 >= 0, s121 <= 0, s122 >= 0, s122 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' >= 0, z - 1 >= 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {G,F}
Previous analysis results are:
  h: runtime: O(1) [0], size: O(1) [0]
  H: runtime: O(n^1) [z + z'], size: O(1) [0]
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]
  G: runtime: INF, size: O(1) [0]
  F: runtime: ?, size: O(n^1) [2 + z']

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

(93) 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:

   g(v0, v1) -> null_g [0]
   h(v0, v1) -> null_h [0]
   f(v0, v1) -> null_f [0]
   F(v0, v1) -> null_F [0]
   G(v0, v1) -> null_G [0]
   H(v0, v1) -> null_H [0]

And the following fresh constants:    null_g, null_h, null_f, null_F, null_G, null_H

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

(94)
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:

   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) [1]
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) [1]
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) [1]
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(0, z0) -> 0 [0]
   h(0, S(z0)) -> h(0, z0) [0]
   h(0, 0) -> 0 [0]
   h(S(z0), z1) -> h(z0, z1) [0]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   g(v0, v1) -> null_g [0]
   h(v0, v1) -> null_h [0]
   f(v0, v1) -> null_f [0]
   F(v0, v1) -> null_F [0]
   G(v0, v1) -> null_G [0]
   H(v0, v1) -> null_H [0]

The TRS has the following type information:
F :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> c:c1:null_F
S :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f
c :: c8:c10:null_H -> c4:c5:null_G -> c:c1:null_F
H :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> c8:c10:null_H
g :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f
f :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f
G :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> c4:c5:null_G
c1 :: c8:c10:null_H -> c:c1:null_F -> c:c1:null_F
c4 :: c8:c10:null_H -> c:c1:null_F -> c4:c5:null_G
c5 :: c8:c10:null_H -> c4:c5:null_G -> c4:c5:null_G
0 :: S:0:null_g:null_h:null_f
c8 :: c8:c10:null_H -> c8:c10:null_H
c10 :: c8:c10:null_H -> c8:c10:null_H
h :: S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f -> S:0:null_g:null_h:null_f
null_g :: S:0:null_g:null_h:null_f
null_h :: S:0:null_g:null_h:null_f
null_f :: S:0:null_g:null_h:null_f
null_F :: c:c1:null_F
null_G :: c4:c5:null_G
null_H :: c8:c10:null_H

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

(95) 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_g => 0
   null_h => 0
   null_f => 0
   null_F => 0
   null_G => 0
   null_H => 0

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

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

   F(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   F(z, z') -{ 1 }-> 1 + H(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)) + G(z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   F(z, z') -{ 1 }-> 1 + H(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)) + F(1 + (1 + (1 + z0)), z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   G(z, z') -{ 1 }-> 1 + H(f(1 + z0, 1 + z1), g(z0, 1 + (1 + (1 + z1)))) + G(z0, 1 + (1 + (1 + z1))) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   G(z, z') -{ 1 }-> 1 + H(f(1 + z0, 1 + z1), g(z0, 1 + (1 + (1 + z1)))) + F(1 + z0, 1 + z1) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   H(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   H(z, z') -{ 1 }-> 1 + H(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   H(z, z') -{ 1 }-> 1 + H(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   f(z, z') -{ 0 }-> h(g(z0, 1 + z1), f(1 + (1 + (1 + z0)), z1)) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   f(z, z') -{ 0 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 0
   f(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   g(z, z') -{ 0 }-> h(f(1 + z0, 1 + z1), g(z0, 1 + (1 + (1 + z1)))) :|: z1 >= 0, z = 1 + z0, z0 >= 0, z' = 1 + z1
   g(z, z') -{ 0 }-> 0 :|: z0 >= 0, z = 0, z' = z0
   g(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   h(z, z') -{ 0 }-> h(z0, z1) :|: z1 >= 0, z = 1 + z0, z' = z1, z0 >= 0
   h(z, z') -{ 0 }-> h(0, z0) :|: z0 >= 0, z' = 1 + z0, z = 0
   h(z, z') -{ 0 }-> 0 :|: z = 0, z' = 0
   h(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   H(0, S(z0)) -> c8(H(0, z0))
We considered the (Usable) Rules:
   f(S(z0), 0) -> 0
   g(0, z0) -> 0
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
And the Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = 0
   POL(G(x_1, x_2)) = 0
   POL(H(x_1, x_2)) = x_2
   POL(S(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(S(z0), z1) -> c10(H(z0, z1))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c4_2, c5_2, c8_1, c10_1


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

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   H(S(z0), z1) -> c10(H(z0, z1))
We considered the (Usable) Rules:
   g(0, z0) -> 0
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
And the Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = 0
   POL(G(x_1, x_2)) = 0
   POL(H(x_1, x_2)) = x_1
   POL(S(x_1)) = [1] + x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
S tuples:
   F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1)))
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c_2, c1_2, c4_2, c5_2, c8_1, c10_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(z0), S(z1)) -> c(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), G(z0, S(z1))) by 
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1)))
S tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)), G(0, S(x1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c1_2, c4_2, c5_2, c8_1, c10_1, c_2


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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
S tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c1_2, c4_2, c5_2, c8_1, c10_1, c_2, c_1


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

(105) 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(x1)) -> c(H(0, f(S(S(S(0))), x1)))
We considered the (Usable) Rules:none
And the Tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = [2]
   POL(G(x_1, x_2)) = [2]
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = [3]
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = [1] + [3]x_1 + [3]x_2
   POL(g(x_1, x_2)) = [3]
   POL(h(x_1, x_2)) = [2] + [2]x_1

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
S tuples:
   F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1))
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F_2, G_2, H_2

Compound Symbols:   c1_2, c4_2, c5_2, c8_1, c10_1, c_2, c_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(z0), S(z1)) -> c1(H(g(z0, S(z1)), f(S(S(S(z0))), z1)), F(S(S(S(z0))), z1)) by 
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
S tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0), F(S(S(S(x0))), 0))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G_2, H_2, F_2

Compound Symbols:   c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
S tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G_2, H_2, F_2

Compound Symbols:   c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1


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

(111) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
We considered the (Usable) Rules:none
And the Tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = [1] + x_2
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
S tuples:
   G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G_2, H_2, F_2

Compound Symbols:   c4_2, c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(z0), S(z1)) -> c4(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) by 
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
S tuples:
   G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G_2, H_2, F_2

Compound Symbols:   c5_2, c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(z0), S(z1)) -> c5(H(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) by 
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
S tuples:
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0), G(0, S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
S tuples:
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
We considered the (Usable) Rules:none
And the Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = x_1
   POL(G(x_1, x_2)) = [1]
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
S tuples:
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) by 
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
S tuples:
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G(0, S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
S tuples:
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(125) 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(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
We considered the (Usable) Rules:none
And the Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = [1] + x_2
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
S tuples:
   F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(x0), S(0)) -> c(H(g(x0, S(0)), 0), G(x0, S(0))) by 
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(0), S(0)) -> c(H(0, 0), G(0, S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(0), S(0)) -> c(H(0, 0), G(0, S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
S tuples:
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(0), S(0)) -> c(H(0, 0), G(0, S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(129) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(S(0), S(0)) -> c(H(0, 0), G(0, S(0)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
S tuples:
   F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_2, c_1, c1_2, c1_1, c4_2, c5_2, c5_1


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(S(z0)), S(z1)) -> c(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1))) by 
   F(S(S(x0)), S(S(z1))) -> c(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), G(S(x0), S(S(z1))))
   F(S(S(x0)), S(0)) -> c(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G(S(x0), S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
S tuples:
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1))) by 
   F(S(0), S(S(z1))) -> c(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))))
   F(S(0), S(0)) -> c(H(0, 0))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(0)) -> c(H(0, 0))
S tuples:
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c(H(0, f(S(S(S(0))), x1)))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

(135) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(S(0), S(0)) -> c(H(0, 0))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
S tuples:
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) by 
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
S tuples:
   F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(S(z0)), S(z1)) -> c1(H(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1)) by 
   F(S(S(x0)), S(S(z1))) -> c1(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), F(S(S(S(S(x0)))), S(z1)))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
S tuples:
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), F(S(S(S(S(x0)))), 0))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
S tuples:
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

(143) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
We considered the (Usable) Rules:none
And the Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = [1] + x_2
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
S tuples:
   F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_2, c1_1, c4_2, c5_2, c5_1, c_2, c_1


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(0), S(x1)) -> c1(H(0, f(S(S(S(0))), x1)), F(S(S(S(0))), x1)) by 
   F(S(0), S(S(z1))) -> c1(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))), F(S(S(S(0))), S(z1)))
   F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2


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

(147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(S(0), S(0)) -> c1(H(0, 0), F(S(S(S(0))), 0))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2


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

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0)) by 
   F(S(S(z0)), S(0)) -> c1(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   F(S(0), S(0)) -> c1(H(0, 0))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   F(S(0), S(0)) -> c1(H(0, 0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(0)) -> c1(H(g(x0, S(0)), 0))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2


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

(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(S(0), S(0)) -> c1(H(0, 0))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2


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

(153) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) by 
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1)))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2, c4_1


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

(155) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(0), S(x1)) -> c4(H(f(S(0), S(x1)), 0), F(S(0), S(x1))) by 
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1)))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c4_2, c5_2, c5_1, c_2, c_1, c1_2, c4_1


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(z0), S(z1)) -> c4(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F(S(z0), S(z1))) by 
   G(S(S(z0)), S(x1)) -> c4(H(h(g(S(z0), S(x1)), f(S(S(S(S(z0)))), x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(0), S(x1)) -> c4(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F(S(0), S(x1)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_2, c5_1, c_2, c_1, c1_2, c4_2, c4_1


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

(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1))))) by 
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_2, c5_1, c_2, c_1, c1_2, c4_2, c4_1


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

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(z0), S(z1)) -> c5(H(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), G(z0, S(S(S(z1))))) by 
   G(S(S(z0)), S(x1)) -> c5(H(h(g(S(z0), S(x1)), f(S(S(S(S(z0)))), x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1)))))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1)))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), G(0, S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), G(0, S(S(S(x1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(165) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
We considered the (Usable) Rules:none
And the Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F(x_1, x_2)) = [1] + x_1
   POL(G(x_1, x_2)) = [1] + x_2
   POL(H(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = 0
   POL(h(x_1, x_2)) = 0

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0)) by 
   G(S(0), S(z1)) -> c5(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(S(z1)))) -> c(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), G(x0, S(S(S(z1))))) by F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(171) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(0))) -> c(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G(x0, S(S(0)))) by F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(173) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(z1))) -> c(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), G(x0, S(S(z1)))) by F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1)))) by F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(177) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1)))) by F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(179) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0))) by F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
----------------------------------------

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(S(z0)), S(S(x1))) -> c(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), G(S(z0), S(S(x1))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_2, c_1, c1_2, c4_2, c4_1, c5_2


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

(181) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(x0)), S(S(z1))) -> c(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), G(S(x0), S(S(z1)))) by F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
----------------------------------------

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
K tuples:
   H(0, S(z0)) -> c8(H(0, z0))
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c8_1, c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2


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

(183) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(0, S(z0)) -> c8(H(0, z0)) by 
   H(0, S(S(y0))) -> c8(H(0, S(y0)))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(z0)), S(0)) -> c(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(x0)), S(0)) -> c(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G(S(x0), S(0))) by F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(S(z0))), S(x1)) -> c(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), G(S(S(z0)), S(x1))) by F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(0)), S(x1)) -> c(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G(S(0), S(x1))) by F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(0), S(S(z1))) -> c(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1)))) by F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
----------------------------------------

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1))))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(S(z1)))) -> c1(H(g(x0, S(S(S(z1)))), h(g(S(S(x0)), S(S(z1))), h(g(S(S(S(S(x0)))), S(z1)), f(S(S(S(S(S(S(S(x0))))))), z1)))), F(S(S(S(x0))), S(S(z1)))) by F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(0))) -> c1(H(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), F(S(S(S(x0))), S(0))) by F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(197) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(x0), S(S(z1))) -> c1(H(g(x0, S(S(z1))), h(h(f(S(S(x0)), S(z1)), g(S(x0), S(S(S(z1))))), f(S(S(S(S(S(x0))))), z1))), F(S(S(S(x0))), S(z1))) by F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
----------------------------------------

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(199) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1))) by F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1))) by F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
----------------------------------------

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(S(z0)), S(S(x1))) -> c1(H(h(f(S(z0), S(S(x1))), g(z0, S(S(S(S(x1)))))), h(g(S(S(S(z0))), S(x1)), f(S(S(S(S(S(S(z0)))))), x1))), F(S(S(S(S(z0)))), S(x1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(203) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(x0)), S(S(z1))) -> c1(H(h(f(S(x0), S(S(z1))), g(x0, S(S(S(S(z1)))))), h(g(S(S(S(x0))), S(z1)), f(S(S(S(S(S(S(x0)))))), z1))), F(S(S(S(S(x0)))), S(z1))) by F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
----------------------------------------

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(S(z0))), S(x1)) -> c1(H(h(f(S(S(z0)), S(x1)), h(f(S(z0), S(S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), f(S(S(S(S(S(z0))))), x1)), F(S(S(S(S(S(z0))))), x1)) by F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(0)), S(x1)) -> c1(H(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), F(S(S(S(S(0)))), x1)) by F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
----------------------------------------

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(209) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0)) by F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
----------------------------------------

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(0), S(S(x1))) -> c1(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F(S(S(S(0))), S(x1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(211) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(0), S(S(z1))) -> c1(H(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))), F(S(S(S(0))), S(z1))) by F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
----------------------------------------

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(213) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(S(S(z0)), S(0)) -> c1(H(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0)) by F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
----------------------------------------

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1)))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(215) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(S(S(S(z0))), S(x1)) -> c4(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), F(S(S(S(z0))), S(x1))) by G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
----------------------------------------

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1)))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(217) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(S(S(0)), S(x1)) -> c4(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F(S(S(0)), S(x1))) by G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0)))
----------------------------------------

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
   G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1)))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
   G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1


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

(219) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(S(S(z0)), S(x1)) -> c4(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), F(S(S(z0)), S(x1))) by G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1)))
----------------------------------------

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1)))))
   g(0, z0) -> 0
   h(0, S(z0)) -> h(0, z0)
   h(0, 0) -> 0
   h(S(z0), z1) -> h(z0, z1)
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1))
   f(S(z0), 0) -> 0
Tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(0), S(S(z0))) -> c(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   F(S(S(z0)), S(0)) -> c1(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
   G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0)))
   G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1)))
S tuples:
   F(S(x0), S(x1)) -> c1(F(S(S(S(x0))), x1))
   G(S(x0), S(x1)) -> c5(G(x0, S(S(S(x1)))))
   F(S(x0), S(S(x1))) -> c(G(x0, S(S(x1))))
   F(S(x0), S(0)) -> c(G(x0, S(0)))
   F(S(S(z0)), S(z1)) -> c(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), G(S(z0), S(z1)))
   F(S(x0), S(S(x1))) -> c1(F(S(S(S(x0))), S(x1)))
   F(S(S(z0)), S(z1)) -> c1(H(h(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), F(S(S(S(S(z0)))), z1))
   F(S(S(x0)), S(x1)) -> c1(F(S(S(S(S(x0)))), x1))
   F(S(0), S(x0)) -> c1(F(S(S(S(0))), x0))
   G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1)))
   G(S(S(x0)), S(x1)) -> c4(F(S(S(x0)), S(x1)))
   G(S(0), S(z1)) -> c4(H(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F(S(0), S(z1)))
   G(S(0), S(x0)) -> c4(F(S(0), S(x0)))
   G(S(x0), S(S(z1))) -> c4(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), F(S(x0), S(S(z1))))
   G(S(x0), S(0)) -> c4(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F(S(x0), S(0)))
   G(S(S(z0)), S(z1)) -> c4(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), F(S(S(z0)), S(z1)))
   G(S(0), S(x1)) -> c4(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F(S(0), S(x1)))
   G(S(x0), S(x1)) -> c4(F(S(x0), S(x1)))
   G(S(S(S(z0))), S(x1)) -> c5(H(f(S(S(S(z0))), S(x1)), h(f(S(S(z0)), S(S(S(x1)))), h(f(S(z0), S(S(S(S(S(x1)))))), g(z0, S(S(S(S(S(S(S(x1))))))))))), G(S(S(z0)), S(S(S(x1)))))
   G(S(S(0)), S(x1)) -> c5(H(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), G(S(0), S(S(S(x1)))))
   G(S(S(z0)), S(x1)) -> c5(H(f(S(S(z0)), S(x1)), h(h(g(z0, S(S(S(x1)))), f(S(S(S(z0))), S(S(x1)))), g(z0, S(S(S(S(S(x1)))))))), G(S(z0), S(S(S(x1)))))
   G(S(S(x0)), S(z1)) -> c5(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), G(S(x0), S(S(S(z1)))))
   G(S(S(x0)), S(x1)) -> c5(G(S(x0), S(S(S(x1)))))
   G(S(x0), S(S(z1))) -> c5(H(h(g(x0, S(S(z1))), h(g(S(S(x0)), S(z1)), f(S(S(S(S(S(x0))))), z1))), g(x0, S(S(S(S(z1)))))), G(x0, S(S(S(S(z1))))))
   G(S(x0), S(0)) -> c5(H(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), G(x0, S(S(S(0)))))
   G(S(S(z0)), S(z1)) -> c5(H(h(h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), f(S(S(S(S(z0)))), z1)), g(S(z0), S(S(S(z1))))), G(S(z0), S(S(S(z1)))))
   F(S(z0), S(S(S(z1)))) -> c(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), G(z0, S(S(S(z1)))))
   F(S(z0), S(S(0))) -> c(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), G(z0, S(S(0))))
   F(S(z0), S(S(z1))) -> c(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), G(z0, S(S(z1))))
   F(S(S(z0)), S(S(z1))) -> c(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), G(S(z0), S(S(z1))))
   F(S(S(z0)), S(0)) -> c(H(h(h(g(z0, S(0)), f(S(S(S(z0))), 0)), g(z0, S(S(S(0))))), 0), G(S(z0), S(0)))
   F(S(S(S(z0))), S(z1)) -> c(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), G(S(S(z0)), S(z1)))
   F(S(S(0)), S(z0)) -> c(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), G(S(0), S(z0)))
   F(S(z0), S(S(S(z1)))) -> c1(H(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), h(h(f(S(S(S(S(z0)))), S(z1)), g(S(S(S(z0))), S(S(S(z1))))), f(S(S(S(S(S(S(S(z0))))))), z1)))), F(S(S(S(z0))), S(S(z1))))
   F(S(z0), S(S(0))) -> c1(H(g(z0, S(S(0))), h(h(f(S(S(z0)), S(0)), g(S(z0), S(S(S(0))))), 0)), F(S(S(S(z0))), S(0)))
   F(S(z0), S(S(z1))) -> c1(H(g(z0, S(S(z1))), h(h(f(S(S(z0)), S(z1)), h(f(S(z0), S(S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1))), F(S(S(S(z0))), S(z1)))
   F(S(S(z0)), S(S(z1))) -> c1(H(h(f(S(z0), S(S(z1))), g(z0, S(S(S(S(z1)))))), h(h(f(S(S(S(z0))), S(z1)), g(S(S(z0)), S(S(S(z1))))), f(S(S(S(S(S(S(z0)))))), z1))), F(S(S(S(S(z0)))), S(z1)))
   F(S(0), S(S(z0))) -> c1(H(0, h(h(f(S(S(0)), S(z0)), g(S(0), S(S(S(z0))))), f(S(S(S(S(S(0))))), z0))), F(S(S(S(0))), S(z0)))
   F(S(S(S(z0))), S(z1)) -> c1(H(h(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), f(S(S(S(z0))), S(S(z1)))), g(z0, S(S(S(S(S(z1)))))))), f(S(S(S(S(S(z0))))), z1)), F(S(S(S(S(S(z0))))), z1))
   F(S(S(0)), S(z0)) -> c1(H(h(h(g(0, S(z0)), f(S(S(S(0))), z0)), 0), f(S(S(S(S(0)))), z0)), F(S(S(S(S(0)))), z0))
   G(S(S(S(z0))), S(z1)) -> c4(H(f(S(S(S(z0))), S(z1)), h(f(S(S(z0)), S(S(S(z1)))), h(h(g(z0, S(S(S(S(S(z1)))))), f(S(S(S(z0))), S(S(S(S(z1)))))), g(z0, S(S(S(S(S(S(S(z1))))))))))), F(S(S(S(z0))), S(z1)))
   G(S(S(0)), S(z0)) -> c4(H(f(S(S(0)), S(z0)), h(h(g(0, S(S(S(z0)))), f(S(S(S(0))), S(S(z0)))), 0)), F(S(S(0)), S(z0)))
   G(S(S(z0)), S(z1)) -> c4(H(f(S(S(z0)), S(z1)), h(h(g(z0, S(S(S(z1)))), h(g(S(S(z0)), S(S(z1))), f(S(S(S(S(S(z0))))), S(z1)))), g(z0, S(S(S(S(S(z1)))))))), F(S(S(z0)), S(z1)))
K tuples:
   H(S(z0), z1) -> c10(H(z0, z1))
   G(S(0), S(x1)) -> c5(H(f(S(0), S(x1)), 0))
   F(S(0), S(S(x1))) -> c(H(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F(S(S(x0)), S(0)) -> c1(H(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   G(S(0), S(x1)) -> c5(H(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0))
   G(S(0), S(x1)) -> c5(H(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   H(0, S(S(y0))) -> c8(H(0, S(y0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H_2, F_2, G_2

Compound Symbols:   c10_1, c1_1, c5_1, c_1, c_2, c1_2, c4_2, c4_1, c5_2, c8_1