KILLED
proof of /export/starexec/sandbox2/benchmark/theBenchmark.trs
# AProVE Commit ID: c69e44bd14796315568835c1ffa2502984884775 mhark 20210624 unpublished


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

(0) CpxRelTRS
(1) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) typed CpxTrs
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
(9) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 1 ms]
(10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 5 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 418 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 376 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 653 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 2718 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 4 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 26.2 s]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 5410 ms]
        (46) CpxRNTS
    (47) CompletionProof [UPPER BOUND(ID), 0 ms]
    (48) CpxTypedWeightedCompleteTrs
        (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
(51) CpxTrsToCdtProof [UPPER BOUND(ID), 2 ms]
(52) CdtProblem
    (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 99 ms]
    (58) CdtProblem
    (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 75 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 141 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 85 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 148 ms]
    (130) CdtProblem
    (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 163 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (146) CdtProblem
    (147) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms]
    (150) CdtProblem
    (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 241 ms]
    (156) CdtProblem
    (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 325 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (200) CdtProblem
    (201) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms]
    (204) CdtProblem
    (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 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


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

(0)
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
----------------------------------------

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

(2)
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))
   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)

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

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

(4)
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
----------------------------------------

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

(6)
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

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

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

(8)
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
----------------------------------------

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

(10)
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]
   F(S(z0), 0) -> c2 [1]
   F(0, z0) -> c3 [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]
   G(S(z0), 0) -> c6 [1]
   G(0, z0) -> c7 [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(0, 0) -> c9 [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   f(0, z0) -> 0 [0]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(S(z0), 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]

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

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

(12)
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]
   F(S(z0), 0) -> c2 [1]
   F(0, z0) -> c3 [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]
   G(S(z0), 0) -> c6 [1]
   G(0, z0) -> c7 [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(0, 0) -> c9 [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   f(0, z0) -> 0 [0]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(S(z0), 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]

The TRS has the following type information:
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

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

(13) 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:

   f_2
   g_2
   h_2

Due to the following rules being added:

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

And the following fresh constants: none

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

(14)
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]
   F(S(z0), 0) -> c2 [1]
   F(0, z0) -> c3 [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]
   G(S(z0), 0) -> c6 [1]
   G(0, z0) -> c7 [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(0, 0) -> c9 [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   f(0, z0) -> 0 [0]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(S(z0), 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(v0, v1) -> 0 [0]
   g(v0, v1) -> 0 [0]
   h(v0, v1) -> 0 [0]

The TRS has the following type information:
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

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

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

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

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

   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]
   F(S(z0), 0) -> c2 [1]
   F(0, z0) -> c3 [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]
   G(S(z0), 0) -> c6 [1]
   G(0, z0) -> c7 [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(0, 0) -> c9 [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   f(S(S(z05)), S(S(z15))) -> h(h(f(S(z05), S(S(z15))), g(z05, S(S(S(S(z15)))))), h(g(S(S(S(z05))), S(z15)), f(S(S(S(S(S(S(z05)))))), z15))) [0]
   f(S(S(z05)), S(0)) -> h(h(f(S(z05), S(0)), g(z05, S(S(S(0))))), 0) [0]
   f(S(S(z05)), S(z1)) -> h(h(f(S(z05), S(z1)), g(z05, 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]
   f(0, z0) -> 0 [0]
   g(S(S(z06)), S(z1)) -> h(h(g(S(z06), S(z1)), f(S(S(S(S(z06)))), z1)), h(f(S(z06), S(S(S(z1)))), g(z06, 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(z07)), S(z1)) -> h(0, h(f(S(z07), S(S(S(z1)))), g(z07, 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(S(z0), 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(v0, v1) -> 0 [0]
   g(v0, v1) -> 0 [0]
   h(v0, v1) -> 0 [0]

The TRS has the following type information:
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

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

(17) 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
   c2 => 0
   c3 => 1
   c6 => 0
   c7 => 1
   c9 => 0

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

(18)
Obligation:
Complexity RNTS consisting of the following rules:

   F(z, z') -{ 1 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   F(z, z') -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 0
   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 :|: z0 >= 0, z = 0, z' = z0
   G(z, z') -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 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 }-> 0 :|: z = 0, z' = 0
   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 + z05, 1 + z1), g(z05, 1 + (1 + (1 + z1)))), 0) :|: z1 >= 0, z05 >= 0, z' = 1 + z1, z = 1 + (1 + z05)
   f(z, z') -{ 0 }-> h(h(f(1 + z05, 1 + 0), g(z05, 1 + (1 + (1 + 0)))), 0) :|: z' = 1 + 0, z05 >= 0, z = 1 + (1 + z05)
   f(z, z') -{ 0 }-> h(h(f(1 + z05, 1 + (1 + z15)), g(z05, 1 + (1 + (1 + (1 + z15))))), h(g(1 + (1 + (1 + z05)), 1 + z15), f(1 + (1 + (1 + (1 + (1 + (1 + z05))))), z15))) :|: z' = 1 + (1 + z15), z15 >= 0, z05 >= 0, z = 1 + (1 + z05)
   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 :|: z0 >= 0, z = 0, z' = z0
   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 + z06, 1 + z1), f(1 + (1 + (1 + (1 + z06))), z1)), 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, h(f(1 + z07, 1 + (1 + (1 + z1))), g(z07, 1 + (1 + (1 + (1 + (1 + z1))))))) :|: z1 >= 0, z = 1 + (1 + z07), z07 >= 0, 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 :|: z = 1 + z0, z0 >= 0, z' = 0
   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


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(19) SimplificationProof (BOTH BOUNDS(ID, ID))
Simplified the RNTS by moving equalities from the constraints into the right-hand sides.
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(20)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

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

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

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

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

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

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(25) 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

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

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

   F(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   F(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 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
   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 :|: z' >= 0, z = 0
   G(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 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 }-> 0 :|: z = 0, z' = 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' = 1 + 0, z - 2 >= 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
   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 - 1 >= 0, z' = 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]

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(27) 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

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

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

   F(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   F(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 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
   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 :|: z' >= 0, z = 0
   G(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 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 }-> 0 :|: z = 0, z' = 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' = 1 + 0, z - 2 >= 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
   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 - 1 >= 0, z' = 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]

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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(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'

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

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

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(35) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(36)
Obligation:
Complexity RNTS consisting of the following rules:

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

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

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

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

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

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

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

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

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

   F(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   F(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   F(z, z') -{ 2 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s22 + s25, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= s29 + 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 <= s33 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 + s37, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s41, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s45 + s48, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s52 + 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s56 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 + s60, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s64, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   G(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   G(z, z') -{ 2 + 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 + s102, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 <= s68 + s71, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= s75 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s79 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 + s83, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s87 + s90, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s94 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s98 + 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   H(z, z') -{ 1 + z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0 + (z' - 1), z' - 1 >= 0, z = 0
   H(z, z') -{ 1 + z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= z - 1 + z', z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(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' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(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
   f(z, z') -{ 0 }-> s126 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, 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
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s133 :|: s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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) [1 + z + z'], size: O(n^1) [z + z']
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   F(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   F(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   F(z, z') -{ 2 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s22 + s25, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= s29 + 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 <= s33 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 + s37, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s41, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s45 + s48, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s52 + 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s56 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 + s60, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s64, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   G(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   G(z, z') -{ 2 + 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 + s102, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 <= s68 + s71, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= s75 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s79 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 + s83, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s87 + s90, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s94 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s98 + 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   H(z, z') -{ 1 + z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0 + (z' - 1), z' - 1 >= 0, z = 0
   H(z, z') -{ 1 + z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= z - 1 + z', z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(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' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(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
   f(z, z') -{ 0 }-> s126 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, 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
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s133 :|: s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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) [1 + z + z'], size: O(n^1) [z + z']
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]
  G: runtime: ?, size: INF
  F: runtime: ?, size: INF

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

(45) 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: ?

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

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

   F(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   F(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   F(z, z') -{ 2 }-> 1 + s10 + F(1 + (1 + (1 + 0)), 0) :|: s10 >= 0, s10 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 }-> 1 + s11 + F(1 + (1 + (1 + 0)), z' - 1) :|: s11 >= 0, s11 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s12 + F(1 + (1 + (1 + (z - 1))), 0) :|: s12 >= 0, s12 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s13 + F(1 + (1 + (1 + (z - 1))), z' - 1) :|: s13 >= 0, s13 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s22 + s25, z - 2 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 + s29 }-> 1 + s30 + G(1 + (z - 2), 1 + 0) :|: s27 >= 0, s27 <= 0, s28 >= 0, s28 <= 0, s29 >= 0, s29 <= 0, s30 >= 0, s30 <= s29 + 0, z - 2 >= 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 <= s33 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 + s37, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s41, z' - 2 >= 0, z - 1 >= 0
   F(z, z') -{ 2 + 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 <= s45 + s48, z' - 2 >= 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s52 + 0, z' = 1 + 0, z - 2 >= 0
   F(z, z') -{ 2 + 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 <= s56 + 0, z' - 1 >= 0, z - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s6 + G(0, 1 + 0) :|: s6 >= 0, s6 <= 0 + 0, z = 1 + 0, z' = 1 + 0
   F(z, z') -{ 2 + 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 + s60, z = 1 + 0, z' - 2 >= 0
   F(z, z') -{ 2 + 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 + s64, z - 1 >= 0, z' - 2 >= 0
   F(z, z') -{ 2 }-> 1 + s7 + G(0, 1 + (z' - 1)) :|: s7 >= 0, s7 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   F(z, z') -{ 2 }-> 1 + s8 + G(z - 1, 1 + 0) :|: s8 >= 0, s8 <= 0 + 0, z' = 1 + 0, z - 1 >= 0
   F(z, z') -{ 2 }-> 1 + s9 + G(z - 1, 1 + (z' - 1)) :|: s9 >= 0, s9 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 1 }-> 1 :|: z' >= 0, z = 0
   G(z, z') -{ 1 }-> 0 :|: z - 1 >= 0, z' = 0
   G(z, z') -{ 2 + 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 + s102, z - 2 >= 0, z' - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s14 + F(1 + 0, 1 + (z' - 1)) :|: s14 >= 0, s14 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s15 + F(1 + (z - 1), 1 + (z' - 1)) :|: s15 >= 0, s15 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 }-> 1 + s16 + G(0, 1 + (1 + (1 + (z' - 1)))) :|: s16 >= 0, s16 <= 0 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 }-> 1 + s17 + G(z - 1, 1 + (1 + (1 + (z' - 1)))) :|: s17 >= 0, s17 <= 0 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 <= s68 + s71, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + s75 }-> 1 + s76 + F(1 + 0, 1 + (z' - 1)) :|: s73 >= 0, s73 <= 0, s74 >= 0, s74 <= 0, s75 >= 0, s75 <= 0, s76 >= 0, s76 <= s75 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s79 + 0, z' - 1 >= 0, z - 1 >= 0
   G(z, z') -{ 2 + 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 + s83, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s87 + s90, z' - 1 >= 0, z - 2 >= 0
   G(z, z') -{ 2 + 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 <= s94 + 0, z' - 1 >= 0, z = 1 + 0
   G(z, z') -{ 2 + 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 <= s98 + 0, z' - 1 >= 0, z - 1 >= 0
   H(z, z') -{ 1 }-> 0 :|: z = 0, z' = 0
   H(z, z') -{ 1 + z' }-> 1 + s18 :|: s18 >= 0, s18 <= 0 + (z' - 1), z' - 1 >= 0, z = 0
   H(z, z') -{ 1 + z + z' }-> 1 + s19 :|: s19 >= 0, s19 <= z - 1 + z', z' >= 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s :|: s >= 0, s <= 0, z = 1 + 0, z' = 1 + 0
   f(z, z') -{ 0 }-> s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z = 1 + 0
   f(z, z') -{ 0 }-> s'' :|: s'' >= 0, s'' <= 0, z' = 1 + 0, z - 1 >= 0
   f(z, z') -{ 0 }-> s1 :|: s1 >= 0, s1 <= 0, z' - 1 >= 0, z - 1 >= 0
   f(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' - 2 >= 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s114 :|: s111 >= 0, s111 <= 0, s112 >= 0, s112 <= 0, s113 >= 0, s113 <= 0, s114 >= 0, s114 <= 0, z' = 1 + 0, z - 2 >= 0
   f(z, z') -{ 0 }-> s118 :|: s115 >= 0, s115 <= 0, s116 >= 0, s116 <= 0, s117 >= 0, s117 <= 0, s118 >= 0, s118 <= 0, z' - 1 >= 0, z - 2 >= 0
   f(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
   f(z, z') -{ 0 }-> s126 :|: s123 >= 0, s123 <= 0, s124 >= 0, s124 <= 0, s125 >= 0, s125 <= 0, s126 >= 0, s126 <= 0, 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
   f(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   g(z, z') -{ 0 }-> s133 :|: s127 >= 0, s127 <= 0, s128 >= 0, s128 <= 0, s129 >= 0, s129 <= 0, s130 >= 0, s130 <= 0, s131 >= 0, s131 <= 0, s132 >= 0, s132 <= 0, s133 >= 0, s133 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s137 :|: s134 >= 0, s134 <= 0, s135 >= 0, s135 <= 0, s136 >= 0, s136 <= 0, s137 >= 0, s137 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s141 :|: s138 >= 0, s138 <= 0, s139 >= 0, s139 <= 0, s140 >= 0, s140 <= 0, s141 >= 0, s141 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> s145 :|: s142 >= 0, s142 <= 0, s143 >= 0, s143 <= 0, s144 >= 0, s144 <= 0, s145 >= 0, s145 <= 0, z' - 1 >= 0, z - 2 >= 0
   g(z, z') -{ 0 }-> s2 :|: s2 >= 0, s2 <= 0, z' - 1 >= 0, z = 1 + 0
   g(z, z') -{ 0 }-> s3 :|: s3 >= 0, s3 <= 0, z' - 1 >= 0, z - 1 >= 0
   g(z, z') -{ 0 }-> 0 :|: z - 1 >= 0, z' = 0
   g(z, z') -{ 0 }-> 0 :|: z' >= 0, z = 0
   g(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   h(z, z') -{ 0 }-> s4 :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 0
   h(z, z') -{ 0 }-> s5 :|: s5 >= 0, s5 <= 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) [1 + z + z'], size: O(n^1) [z + z']
  f: runtime: O(1) [0], size: O(1) [0]
  g: runtime: O(1) [0], size: O(1) [0]
  G: runtime: INF, size: INF
  F: runtime: ?, size: INF

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

(47) 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:

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

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

(48)
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]
   F(S(z0), 0) -> c2 [1]
   F(0, z0) -> c3 [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]
   G(S(z0), 0) -> c6 [1]
   G(0, z0) -> c7 [1]
   H(0, S(z0)) -> c8(H(0, z0)) [1]
   H(0, 0) -> c9 [1]
   H(S(z0), z1) -> c10(H(z0, z1)) [1]
   f(S(z0), S(z1)) -> h(g(z0, S(z1)), f(S(S(S(z0))), z1)) [0]
   f(S(z0), 0) -> 0 [0]
   f(0, z0) -> 0 [0]
   g(S(z0), S(z1)) -> h(f(S(z0), S(z1)), g(z0, S(S(S(z1))))) [0]
   g(S(z0), 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(v0, v1) -> null_f [0]
   g(v0, v1) -> null_g [0]
   h(v0, v1) -> null_h [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_f:null_g:null_h -> S:0:null_f:null_g:null_h -> c:c1:c2:c3:null_F
S :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h
c :: c8:c9:c10:null_H -> c4:c5:c6:c7:null_G -> c:c1:c2:c3:null_F
H :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h -> c8:c9:c10:null_H
g :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h
f :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h
G :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h -> c4:c5:c6:c7:null_G
c1 :: c8:c9:c10:null_H -> c:c1:c2:c3:null_F -> c:c1:c2:c3:null_F
0 :: S:0:null_f:null_g:null_h
c2 :: c:c1:c2:c3:null_F
c3 :: c:c1:c2:c3:null_F
c4 :: c8:c9:c10:null_H -> c:c1:c2:c3:null_F -> c4:c5:c6:c7:null_G
c5 :: c8:c9:c10:null_H -> c4:c5:c6:c7:null_G -> c4:c5:c6:c7:null_G
c6 :: c4:c5:c6:c7:null_G
c7 :: c4:c5:c6:c7:null_G
c8 :: c8:c9:c10:null_H -> c8:c9:c10:null_H
c9 :: c8:c9:c10:null_H
c10 :: c8:c9:c10:null_H -> c8:c9:c10:null_H
h :: S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h -> S:0:null_f:null_g:null_h
null_f :: S:0:null_f:null_g:null_h
null_g :: S:0:null_f:null_g:null_h
null_h :: S:0:null_f:null_g:null_h
null_F :: c:c1:c2:c3:null_F
null_G :: c4:c5:c6:c7:null_G
null_H :: c8:c9:c10:null_H

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

(49) 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
   c2 => 0
   c3 => 1
   c6 => 0
   c7 => 1
   c9 => 0
   null_f => 0
   null_g => 0
   null_h => 0
   null_F => 0
   null_G => 0
   null_H => 0

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

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

   F(z, z') -{ 1 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   F(z, z') -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 0
   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') -{ 1 }-> 1 :|: z0 >= 0, z = 0, z' = z0
   G(z, z') -{ 1 }-> 0 :|: z = 1 + z0, z0 >= 0, z' = 0
   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') -{ 1 }-> 0 :|: z = 0, z' = 0
   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 :|: z0 >= 0, z = 0, z' = z0
   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 :|: z = 1 + z0, z0 >= 0, z' = 0
   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.

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

(51) CpxTrsToCdtProof (UPPER BOUND(ID))
Converted Cpx (relative) TRS to CDT
----------------------------------------

(52)
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)
   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))
Tuples:
   F'(S(z0), S(z1)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   F'(S(z0), 0) -> c12
   F'(0, z0) -> c13
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(z0), 0) -> c15
   G'(0, z0) -> c16
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(0, 0) -> c18
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   F''(S(z0), 0) -> c22
   F''(0, z0) -> c23
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(z0), 0) -> c26
   G''(0, z0) -> c27
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(0, 0) -> c29
   H''(S(z0), z1) -> c30(H''(z0, z1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   F''(S(z0), 0) -> c22
   F''(0, z0) -> c23
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(z0), 0) -> c26
   G''(0, z0) -> c27
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(0, 0) -> c29
   H''(S(z0), z1) -> c30(H''(z0, z1))
K tuples:none
Defined Rule Symbols:   F_2, G_2, H_2, f_2, g_2, h_2

Defined Pair Symbols:   F'_2, G'_2, H'_2, F''_2, G''_2, H''_2

Compound Symbols:   c11_3, c12, c13, c14_3, c15, c16, c17_1, c18, c19_1, c20_4, c21_4, c22, c23, c24_4, c25_4, c26, c27, c28_1, c29, c30_1


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

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 10 trailing nodes:
   H''(0, 0) -> c29
   F''(S(z0), 0) -> c22
   F'(S(z0), 0) -> c12
   G''(0, z0) -> c27
   G'(S(z0), 0) -> c15
   F''(0, z0) -> c23
   G'(0, z0) -> c16
   H'(0, 0) -> c18
   F'(0, z0) -> c13
   G''(S(z0), 0) -> c26

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

(54)
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)
   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))
Tuples:
   F'(S(z0), S(z1)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
K tuples:none
Defined Rule Symbols:   F_2, G_2, H_2, f_2, g_2, h_2

Defined Pair Symbols:   F'_2, G'_2, H'_2, F''_2, G''_2, H''_2

Compound Symbols:   c11_3, c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1


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

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

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

(56)
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)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
K tuples:none
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F'_2, G'_2, H'_2, F''_2, G''_2, H''_2

Compound Symbols:   c11_3, c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1


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

(57) 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)) -> c28(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)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(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(F''(x_1, x_2)) = 0
   POL(G'(x_1, x_2)) = 0
   POL(G''(x_1, x_2)) = 0
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = x_2
   POL(S(x_1)) = [1] + x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(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

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

(58)
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)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(S(z0), z1) -> c30(H''(z0, z1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F'_2, G'_2, H'_2, F''_2, G''_2, H''_2

Compound Symbols:   c11_3, c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1


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

(59) 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) -> c30(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)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(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(F''(x_1, x_2)) = 0
   POL(G'(x_1, x_2)) = 0
   POL(G''(x_1, x_2)) = 0
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = x_1
   POL(S(x_1)) = [1] + x_1
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(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

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

(60)
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)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1))
   G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   F'_2, G'_2, H'_2, F''_2, G''_2, H''_2

Compound Symbols:   c11_3, c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F'(S(z0), S(z1)) -> c11(H'(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1)) by 
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1))

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

(62)
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)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G'_2, H'_2, F''_2, G''_2, H''_2, F'_2

Compound Symbols:   c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3


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

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

(64)
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)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   G'_2, H'_2, F''_2, G''_2, H''_2, F'_2

Compound Symbols:   c14_3, c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G'(S(z0), S(z1)) -> c14(H'(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1))))) by 
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))

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

(66)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3


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

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

(68)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
S tuples:
   F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1)))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c20_4, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(z0), S(z1)) -> c20(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), G''(z0, S(z1))) by 
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), G''(x0, S(0)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(0), S(x1)) -> c20(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), G''(0, S(x1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))

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

(70)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), G''(x0, S(0)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(0), S(x1)) -> c20(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), G''(0, S(x1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), G''(x0, S(0)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(0), S(x1)) -> c20(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), G''(0, S(x1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1


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

(71) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(72)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c20(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c20(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c20_2


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

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

(74)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1


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

(75) 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(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(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)) = 0
   POL(F''(x_1, x_2)) = x_1
   POL(G'(x_1, x_2)) = 0
   POL(G''(x_1, x_2)) = [1]
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(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

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

(76)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
S tuples:
   F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, F''_2, G''_2, H''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c21_4, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(z0), S(z1)) -> c21(H''(g(z0, S(z1)), f(S(S(S(z0))), z1)), G'(z0, S(z1)), F'(S(S(S(z0))), z1), F''(S(S(S(z0))), z1)) by 
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(x0), S(0)) -> c21(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), F''(S(S(S(x0))), 0))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))

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

(78)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(x0), S(0)) -> c21(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), F''(S(S(S(x0))), 0))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
S tuples:
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(x0), S(0)) -> c21(H''(g(x0, S(0)), 0), G'(x0, S(0)), F'(S(S(S(x0))), 0), F''(S(S(S(x0))), 0))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), G'(0, S(x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2


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

(79) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(80)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c21(H''(g(x0, S(0)), 0), G'(x0, S(0)))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
S tuples:
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c21(H''(g(x0, S(0)), 0), G'(x0, S(0)))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3


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

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

(82)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
S tuples:
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1


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

(83) 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))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F'(x_1, x_2)) = [3]x_1
   POL(F''(x_1, x_2)) = [1] + [3]x_1
   POL(G'(x_1, x_2)) = [3]x_2
   POL(G''(x_1, x_2)) = [1] + [3]x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(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

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

(84)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
S tuples:
   G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c24_4, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(z0), S(z1)) -> c24(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1))) by 
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))

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

(86)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
S tuples:
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1


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

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

(88)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
S tuples:
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3


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

(89) 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(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F'(x_1, x_2)) = 0
   POL(F''(x_1, x_2)) = x_1
   POL(G'(x_1, x_2)) = 0
   POL(G''(x_1, x_2)) = [1]
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(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

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

(90)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
S tuples:
   G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, G''_2, H''_2, F'_2, G'_2, F''_2

Compound Symbols:   c17_1, c19_1, c25_4, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(z0), S(z1)) -> c25(H''(f(S(z0), S(z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1))))) by 
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))

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

(92)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2


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

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

(94)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2


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

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

(96)
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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1


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

(97) 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)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F'(x_1, x_2)) = x_1
   POL(F''(x_1, x_2)) = [1] + x_1
   POL(G'(x_1, x_2)) = x_2
   POL(G''(x_1, x_2)) = [1] + x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(f(x_1, x_2)) = 0
   POL(g(x_1, x_2)) = x_1
   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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1)))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F'(S(x0), S(S(z1))) -> c11(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(S(S(x0))), S(z1))) by 
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(0), S(S(x1))) -> c11(H'(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(0), S(S(x1))) -> c11(H'(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_1


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

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(H'(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F'(S(S(S(0))), S(x1)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_3, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F'(S(S(z0)), S(z1)) -> c11(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(S(S(S(z0)))), z1)) by 
   F'(S(S(x0)), S(S(z1))) -> c11(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(S(S(x0)))), S(z1)))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0))
   F'(S(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0))
   F'(S(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1


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

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0)))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F'(S(x0), S(0)) -> c11(H'(g(x0, S(0)), 0), G'(x0, S(0))) by 
   F'(S(S(z0)), S(0)) -> c11(H'(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G'(S(z0), S(0)))
   F'(S(0), S(0)) -> c11(H'(0, 0), G'(0, S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(0), S(0)) -> c11(H'(0, 0), G'(0, S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1


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

(109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F'(S(0), S(0)) -> c11(H'(0, 0), G'(0, S(0)))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F'_2, G'_2, F''_2, G''_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c11_2, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F'(S(0), S(x1)) -> c11(H'(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1)) by 
   F'(S(0), S(S(z1))) -> c11(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)) -> c11(H'(0, 0), F'(S(S(S(0))), 0))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(0)) -> c11(H'(0, 0), F'(S(S(S(0))), 0))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, G'_2, F''_2, G''_2, F'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2


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

(113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F'(S(0), S(0)) -> c11(H'(0, 0), F'(S(S(S(0))), 0))

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

(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:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, G'_2, F''_2, G''_2, F'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1))))) by 
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, G'_2, F''_2, G''_2, F'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c14_3, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G'(S(z0), S(z1)) -> c14(H'(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1))))) by 
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, G'_2, F''_2, G''_2, F'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1)))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, G'_2, F''_2, G''_2, F'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c14_2, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G'(S(0), S(x1)) -> c14(H'(f(S(0), S(x1)), 0), F'(S(0), S(x1))) by 
   G'(S(0), S(z1)) -> c14(H'(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F'(S(0), S(z1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(x0), S(S(z1))) -> c20(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(S(S(x0))), S(z1)), G''(x0, S(S(z1)))) by 
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(0), S(S(x1))) -> c20(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), G''(0, S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(0), S(S(x1))) -> c20(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), G''(0, S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
S tuples:
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(0), S(S(x1))) -> c20(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), G''(0, S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c20(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F'(S(S(S(0))), S(x1)))
S tuples:
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c20(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), F'(S(S(S(0))), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_2


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
S tuples:
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c3_1


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

(129) 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))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F'(x_1, x_2)) = x_1
   POL(F''(x_1, x_2)) = [1] + x_1
   POL(G'(x_1, x_2)) = x_2
   POL(G''(x_1, x_2)) = [1] + x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
S tuples:
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_4, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c3_1


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(S(z0)), S(z1)) -> c20(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(S(S(S(z0)))), z1), G''(S(z0), S(z1))) by 
   F''(S(S(x0)), S(S(z1))) -> c20(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(S(S(x0)))), S(z1)), G''(S(x0), S(S(z1))))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), G''(S(x0), S(0)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), G''(S(x0), S(0)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
S tuples:
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), G''(S(x0), S(0)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c3_1


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
S tuples:
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c3_1


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

(135) 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(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(F'(x_1, x_2)) = 0
   POL(F''(x_1, x_2)) = [1]
   POL(G'(x_1, x_2)) = 0
   POL(G''(x_1, x_2)) = x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
S tuples:
   F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c20_3, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c3_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(x0), S(0)) -> c20(H''(g(x0, S(0)), 0), G'(x0, S(0)), G''(x0, S(0))) by 
   F''(S(S(z0)), S(0)) -> c20(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0), G'(S(z0), S(0)), G''(S(z0), S(0)))
   F''(S(0), S(0)) -> c20(H''(0, 0), G'(0, S(0)), G''(0, S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(0), S(0)) -> c20(H''(0, 0), G'(0, S(0)), G''(0, S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(0), S(0)) -> c20(H''(0, 0), G'(0, S(0)), G''(0, S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

(139) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F''(S(0), S(0)) -> c20(H''(0, 0), G'(0, S(0)), G''(0, S(0)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
S tuples:
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

(141) 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))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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))
S tuples:
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(H''(0, f(S(S(S(0))), x1)))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

(143) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F''(S(0), S(0)) -> c(H''(0, 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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(0), S(S(z1))) -> c(H''(0, h(g(S(S(0)), S(z1)), f(S(S(S(S(S(0))))), z1))))
S tuples:
   F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(x0), S(S(z1))) -> c21(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(S(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1))) by 
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), F''(S(S(S(0))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), F''(S(S(S(0))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
S tuples:
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))), G'(0, S(S(x1))), F'(S(S(S(0))), S(x1)), F''(S(S(S(0))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
S tuples:
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_4, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2


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

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(S(z0)), S(z1)) -> c21(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(S(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1)) by 
   F''(S(S(x0)), S(S(z1))) -> c21(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(S(S(x0)))), S(z1)), F''(S(S(S(S(x0)))), S(z1)))
   F''(S(S(x0)), S(0)) -> c21(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), F''(S(S(S(S(x0)))), 0))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(x0)), S(0)) -> c21(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), F''(S(S(S(S(x0)))), 0))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(x0)), S(0)) -> c21(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), F'(S(S(S(S(x0)))), 0), F''(S(S(S(S(x0)))), 0))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c21(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c21(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c4_1


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

(155) 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)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
We considered the (Usable) Rules:none
And the Tuples:
   H'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F''(x_1, x_2)) = [1] + x_1
   POL(G'(x_1, x_2)) = x_2
   POL(G''(x_1, x_2)) = [1] + x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c4(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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c21_3, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c4_1


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F''(S(0), S(x1)) -> c21(H''(0, f(S(S(S(0))), x1)), F'(S(S(S(0))), x1), F''(S(S(S(0))), x1)) by 
   F''(S(0), S(S(z1))) -> c21(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(S(S(0))), S(z1)))
   F''(S(0), S(0)) -> c21(H''(0, 0), F'(S(S(S(0))), 0), F''(S(S(S(0))), 0))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(0)) -> c21(H''(0, 0), F'(S(S(S(0))), 0), F''(S(S(S(0))), 0))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(0)) -> c21(H''(0, 0), F'(S(S(S(0))), 0), F''(S(S(S(0))), 0))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1


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

(159) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F''(S(0), S(0)) -> c21(H''(0, 0), F'(S(S(S(0))), 0), F''(S(S(S(0))), 0))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1


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

(161) 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))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   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))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(H''(g(x0, S(0)), 0))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1


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

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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1


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

(165) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1))) by 
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_4, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(z0), S(z1)) -> c24(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), F''(S(z0), S(z1))) by 
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), F''(S(0), S(x1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c24_3, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(0), S(x1)) -> c24(H''(f(S(0), S(x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1))) by 
   G''(S(0), S(z1)) -> c24(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0), F'(S(0), S(z1)), F''(S(0), S(z1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2


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

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1))))) by 
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_4, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_3


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(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(z0), S(z1)) -> c25(H''(h(g(z0, S(z1)), f(S(S(S(z0))), z1)), g(z0, S(S(S(z1))))), F'(S(z0), S(z1)), G'(z0, S(S(S(z1)))), G''(z0, S(S(S(z1))))) by 
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(0), S(x1)) -> c25(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(0), S(x1)) -> c25(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(0), S(x1)) -> c25(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), G'(0, S(S(S(x1)))), G''(0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c25(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c25(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c25(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3


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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   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)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
Defined Rule Symbols:   g_2, h_2, f_2

Defined Pair Symbols:   H'_2, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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(181) 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(F'(S(0), S(x1)))
   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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   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)) = x_1
   POL(F''(x_1, x_2)) = [1] + x_1
   POL(G'(x_1, x_2)) = x_2
   POL(G''(x_1, x_2)) = [1] + x_2
   POL(H'(x_1, x_2)) = 0
   POL(H''(x_1, x_2)) = 0
   POL(S(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c11(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c14(x_1)) = x_1
   POL(c14(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c17(x_1)) = x_1
   POL(c19(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c20(x_1)) = x_1
   POL(c20(x_1, x_2)) = x_1 + x_2
   POL(c20(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c20(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c21(x_1, x_2)) = x_1 + x_2
   POL(c21(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c21(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c24(x_1)) = x_1
   POL(c24(x_1, x_2)) = x_1 + x_2
   POL(c24(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c24(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c25(x_1, x_2)) = x_1 + x_2
   POL(c25(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c25(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c30(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(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

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

(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)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0)) by 
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))

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

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1))))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(x0), S(S(S(z1)))) -> c11(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(S(S(x0))), S(S(z1)))) by F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
----------------------------------------

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(x0), S(S(0))) -> c11(H'(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0))) by F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
----------------------------------------

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(x0), S(S(z1))) -> c11(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(S(x0))), S(z1))) by F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
----------------------------------------

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(191) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1))) by F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
----------------------------------------

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(0), S(S(x1))) -> c11(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))) -> c11(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)))
----------------------------------------

(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'(0, S(z0)) -> c17(H'(0, z0))
   H'(S(z0), z1) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c17_1, c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1


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

(195) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H'(0, S(z0)) -> c17(H'(0, z0)) by 
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))

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

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(S(z0)), S(S(x1))) -> c11(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(S(S(S(z0)))), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_3, c11_1, c11_2, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(197) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(x0)), S(S(z1))) -> c11(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(S(S(x0)))), S(z1))) by F'(S(S(z0)), S(S(z1))) -> c11(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(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1))
   F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_2, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(199) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(S(z0))), S(x1)) -> c11(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(S(S(S(z0))))), x1)) by F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), 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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_2, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(201) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(0)), S(x1)) -> c11(H'(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1)) by F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), 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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_2, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(203) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(x0)), S(0)) -> c11(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)) -> c11(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)))
----------------------------------------

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(S(x0)), S(0)) -> c11(H'(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_2, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(S(z0)), S(0)) -> c11(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)) -> c11(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)))
----------------------------------------

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(0), S(S(x1))) -> c11(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(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_2, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c17_1


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

(207) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F'(S(0), S(S(z1))) -> c11(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))) -> c11(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)))
----------------------------------------

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1)))))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c11_2, c17_1


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

(209) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G'(S(S(S(z0))), S(x1)) -> c14(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(z0)), S(S(S(x1))))) by G'(S(S(S(z0))), S(z1)) -> c14(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(z0)), S(S(S(z1)))))
----------------------------------------

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)))
   G'(S(S(S(z0))), S(z1)) -> c14(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(z0)), S(S(S(z1)))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c11_2, c17_1


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

(211) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G'(S(S(0)), S(x1)) -> c14(H'(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1))))) by G'(S(S(0)), S(z0)) -> c14(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(0), S(S(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1)))))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)))
   G'(S(S(S(z0))), S(z1)) -> c14(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(z0)), S(S(S(z1)))))
   G'(S(S(0)), S(z0)) -> c14(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(0), S(S(S(z0)))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c11_2, c17_1


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(213) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G'(S(S(z0)), S(x1)) -> c14(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(z0), S(S(S(x1))))) by G'(S(S(z0)), S(z1)) -> c14(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)), G'(S(z0), S(S(S(z1)))))
----------------------------------------

(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) -> c19(H'(z0, z1))
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F'(S(x0), S(S(x1))) -> c11(G'(x0, S(S(x1))))
   F'(S(S(z0)), S(z1)) -> c11(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(S(S(z0)))), z1))
   F'(S(S(x0)), S(x1)) -> c11(F'(S(S(S(S(x0)))), x1))
   F'(S(x0), S(0)) -> c11(G'(x0, S(0)))
   F'(S(0), S(x0)) -> c11(F'(S(S(S(0))), x0))
   G'(S(S(x0)), S(z1)) -> c14(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(x0), S(S(S(z1)))))
   G'(S(S(x0)), S(x1)) -> c14(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))))
   G'(S(x0), S(S(z1))) -> c14(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'(x0, S(S(S(S(z1))))))
   G'(S(x0), S(0)) -> c14(H'(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))))
   G'(S(S(z0)), S(z1)) -> c14(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(z0), S(S(S(z1)))))
   G'(S(x0), S(x1)) -> c14(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))))
   G'(S(0), S(x1)) -> c14(H'(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)))
   G'(S(0), S(x1)) -> c14(H'(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)))
   G'(S(0), S(x0)) -> c14(F'(S(0), S(x0)))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   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(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(0)))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   F''(S(S(z0)), S(0)) -> c1(H''(h(f(S(z0), S(0)), g(z0, S(S(S(0))))), 0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(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''(S(0), S(x1)) -> c5(F'(S(0), S(x1)))
   G''(S(0), S(x1)) -> c5(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))))
   G''(S(0), S(z1)) -> c2(H''(h(g(0, S(z1)), f(S(S(S(0))), z1)), 0))
   F'(S(z0), S(S(S(z1)))) -> c11(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(S(S(z0))), S(S(z1))))
   F'(S(z0), S(S(0))) -> c11(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(S(S(z0))), S(0)))
   F'(S(z0), S(S(z1))) -> c11(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(S(z0))), S(z1)))
   F'(S(S(z0)), S(S(z1))) -> c11(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(S(S(z0)))), S(z1)))
   F'(S(0), S(S(z0))) -> c11(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)))
   H'(0, S(S(y0))) -> c17(H'(0, S(y0)))
   F'(S(S(S(z0))), S(z1)) -> c11(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(S(S(S(z0))))), z1))
   F'(S(S(0)), S(z0)) -> c11(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(S(S(S(0)))), z0))
   F'(S(S(z0)), S(0)) -> c11(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)))
   G'(S(S(S(z0))), S(z1)) -> c14(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(z0)), S(S(S(z1)))))
   G'(S(S(0)), S(z0)) -> c14(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(0), S(S(S(z0)))))
   G'(S(S(z0)), S(z1)) -> c14(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)), G'(S(z0), S(S(S(z1)))))
S tuples:
   F''(S(x0), S(x1)) -> c21(F'(S(S(S(x0))), x1), F''(S(S(S(x0))), x1))
   G''(S(x0), S(x1)) -> c25(G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
   F''(S(x0), S(S(S(z1)))) -> c20(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(S(S(x0))), S(S(z1))), G''(x0, S(S(S(z1)))))
   F''(S(x0), S(S(0))) -> c20(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), G''(x0, S(S(0))))
   F''(S(x0), S(S(z1))) -> c20(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(S(x0))), S(z1)), G''(x0, S(S(z1))))
   F''(S(S(z0)), S(S(x1))) -> c20(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(S(S(S(z0)))), S(x1)), G''(S(z0), S(S(x1))))
   F''(S(x0), S(S(x1))) -> c20(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), G''(x0, S(S(x1))))
   F''(S(S(S(z0))), S(x1)) -> c20(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(S(S(S(z0))))), x1), G''(S(S(z0)), S(x1)))
   F''(S(S(0)), S(x1)) -> c20(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), G''(S(0), S(x1)))
   F''(S(S(z0)), S(z1)) -> c20(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(S(S(z0)))), z1), G''(S(z0), S(z1)))
   F''(S(S(x0)), S(0)) -> c20(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0), G'(S(x0), S(0)), G''(S(x0), S(0)))
   F''(S(x0), S(0)) -> c20(G'(x0, S(0)), G''(x0, S(0)))
   F''(S(x0), S(S(S(z1)))) -> c21(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(S(S(x0))), S(S(z1))), F''(S(S(S(x0))), S(S(z1))))
   F''(S(x0), S(S(0))) -> c21(H''(g(x0, S(S(0))), h(g(S(S(x0)), S(0)), 0)), G'(x0, S(S(0))), F'(S(S(S(x0))), S(0)), F''(S(S(S(x0))), S(0)))
   F''(S(x0), S(S(z1))) -> c21(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(S(x0))), S(z1)), F''(S(S(S(x0))), S(z1)))
   F''(S(S(z0)), S(S(x1))) -> c21(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(S(S(S(z0)))), S(x1)), F''(S(S(S(S(z0)))), S(x1)))
   F''(S(x0), S(S(x1))) -> c21(G'(x0, S(S(x1))), F'(S(S(S(x0))), S(x1)), F''(S(S(S(x0))), S(x1)))
   F''(S(0), S(S(x1))) -> c21(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(S(S(0))), S(x1)))
   F''(S(S(S(z0))), S(x1)) -> c21(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(S(S(S(z0))))), x1), F''(S(S(S(S(S(z0))))), x1))
   F''(S(S(0)), S(x1)) -> c21(H''(h(f(S(0), S(x1)), 0), f(S(S(S(S(0)))), x1)), G'(S(0), S(x1)), F'(S(S(S(S(0)))), x1), F''(S(S(S(S(0)))), x1))
   F''(S(S(z0)), S(z1)) -> c21(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(S(S(z0)))), z1), F''(S(S(S(S(z0)))), z1))
   F''(S(S(x0)), S(x1)) -> c21(F'(S(S(S(S(x0)))), x1), F''(S(S(S(S(x0)))), x1))
   F''(S(0), S(x0)) -> c21(F'(S(S(S(0))), x0), F''(S(S(S(0))), x0))
   G''(S(S(S(z0))), S(x1)) -> c24(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(z0)), S(S(S(x1)))), F''(S(S(S(z0))), S(x1)))
   G''(S(S(0)), S(x1)) -> c24(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), F''(S(S(0)), S(x1)))
   G''(S(S(z0)), S(x1)) -> c24(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(z0), S(S(S(x1)))), F''(S(S(z0)), S(x1)))
   G''(S(S(x0)), S(z1)) -> c24(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(x0), S(S(S(z1)))), F''(S(S(x0)), S(z1)))
   G''(S(S(x0)), S(x1)) -> c24(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), F''(S(S(x0)), S(x1)))
   G''(S(x0), S(S(z1))) -> c24(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'(x0, S(S(S(S(z1))))), F''(S(x0), S(S(z1))))
   G''(S(x0), S(0)) -> c24(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), F''(S(x0), S(0)))
   G''(S(S(z0)), S(z1)) -> c24(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(z0), S(S(S(z1)))), F''(S(S(z0)), S(z1)))
   G''(S(x0), S(x1)) -> c24(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), F''(S(x0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(g(0, S(x1)), f(S(S(S(0))), x1)), 0), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x1)) -> c24(H''(h(0, f(S(S(S(0))), x1)), g(0, S(S(S(x1))))), F'(S(0), S(x1)), F''(S(0), S(x1)))
   G''(S(0), S(x0)) -> c24(F'(S(0), S(x0)), F''(S(0), S(x0)))
   G''(S(S(S(z0))), S(x1)) -> c25(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(z0)), S(S(S(x1)))), G''(S(S(z0)), S(S(S(x1)))))
   G''(S(S(0)), S(x1)) -> c25(H''(f(S(S(0)), S(x1)), h(f(S(0), S(S(S(x1)))), 0)), F'(S(S(0)), S(x1)), G'(S(0), S(S(S(x1)))), G''(S(0), S(S(S(x1)))))
   G''(S(S(z0)), S(x1)) -> c25(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(z0), S(S(S(x1)))), G''(S(z0), S(S(S(x1)))))
   G''(S(S(x0)), S(z1)) -> c25(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(x0), S(S(S(z1)))), G''(S(x0), S(S(S(z1)))))
   G''(S(S(x0)), S(x1)) -> c25(F'(S(S(x0)), S(x1)), G'(S(x0), S(S(S(x1)))), G''(S(x0), S(S(S(x1)))))
   G''(S(x0), S(S(z1))) -> c25(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'(x0, S(S(S(S(z1))))), G''(x0, S(S(S(S(z1))))))
   G''(S(x0), S(0)) -> c25(H''(h(g(x0, S(0)), 0), g(x0, S(S(S(0))))), F'(S(x0), S(0)), G'(x0, S(S(S(0)))), G''(x0, S(S(S(0)))))
   G''(S(S(z0)), S(z1)) -> c25(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(z0), S(S(S(z1)))), G''(S(z0), S(S(S(z1)))))
   G''(S(x0), S(x1)) -> c25(F'(S(x0), S(x1)), G'(x0, S(S(S(x1)))), G''(x0, S(S(S(x1)))))
K tuples:
   H''(0, S(z0)) -> c28(H''(0, z0))
   H''(S(z0), z1) -> c30(H''(z0, z1))
   F''(S(x0), S(x1)) -> c20(F'(S(S(S(x0))), x1))
   F''(S(0), S(x1)) -> c(F'(S(S(S(0))), x1))
   F''(S(x0), S(0)) -> c1(G'(x0, S(0)))
   G''(S(x0), S(x1)) -> c24(G'(x0, S(S(S(x1)))))
   G''(S(0), S(x1)) -> c2(H''(f(S(0), S(x1)), 0))
   G''(S(0), S(x1)) -> c2(F'(S(0), S(x1)))
   F''(S(0), S(S(x1))) -> c3(H''(0, h(g(S(S(0)), S(x1)), f(S(S(S(S(S(0))))), x1))))
   F''(S(0), S(S(x1))) -> c3(F'(S(S(S(0))), S(x1)))
   F''(S(S(x0)), S(x1)) -> c20(F'(S(S(S(S(x0)))), x1))
   F''(S(S(x0)), S(0)) -> c4(H''(h(f(S(x0), S(0)), g(x0, S(S(S(0))))), 0))
   F''(S(S(x0)), S(0)) -> c4(G'(S(x0), S(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(F'(S(0), S(x1)))
   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, H''_2, F''_2, G''_2, F'_2, G'_2

Compound Symbols:   c19_1, c28_1, c30_1, c20_1, c_1, c21_2, c1_1, c24_1, c25_2, c2_1, c11_1, c11_3, c14_3, c14_2, c14_1, c20_4, c20_3, c3_1, c20_2, c21_4, c21_3, c4_1, c24_4, c24_3, c24_2, c25_4, c25_3, c5_1, c11_2, c17_1