KILLED
proof of input_dvCxkb8j5l.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 319 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 60 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 1394 ms]
        (30) CpxRNTS
    (31) CompletionProof [UPPER BOUND(ID), 0 ms]
    (32) CpxTypedWeightedCompleteTrs
        (33) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
(35) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(36) CdtProblem
    (37) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 178 ms]
    (42) CdtProblem
    (43) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 53 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 55 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 48 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 79 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 7 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 122 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 153 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem


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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   g_2
   f_2
   h_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(10)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

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

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

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

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

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

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

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

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

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

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

   0 => 0

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

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

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


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

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

(16)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

   { h }
   { f, g }

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

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

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

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

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

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

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

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

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

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

(21) 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

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

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

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

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

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

(23) 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'

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

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

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

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

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

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

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

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

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

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

(27) 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

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

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

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

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(31) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(32)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

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

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

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

(33) 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

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(36)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

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

(38)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

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

(40)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

Polynomial interpretation :

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

Polynomial interpretation :

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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

Polynomial interpretation :

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   F_2, G_2, H_2

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


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

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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   G_2, H_2, F_2

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


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   G_2, H_2, F_2

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


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

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

Polynomial interpretation :

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

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

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

Defined Pair Symbols:   G_2, H_2, F_2

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


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

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

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

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

Defined Pair Symbols:   G_2, H_2, F_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

Polynomial interpretation :

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

Polynomial interpretation :

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

(77) 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))

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

Polynomial interpretation :

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

(93) 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))

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

Polynomial interpretation :

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(116)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(118)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(120)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(122)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(124)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(126)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(130)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(132)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(134)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(136)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(138)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(140)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(142)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(144)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(146)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(148)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(150)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(152)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(154)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(156)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(158)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(160)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

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

(162)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(S(S(x0)), S(z1)) -> c4(H(h(g(S(x0), S(z1)), f(S(S(S(S(x0)))), z1)), h(f(S(x0), S(S(S(z1)))), g(x0, S(S(S(S(S(z1)))))))), F(S(S(x0)), S(z1))) by G(S(S(z0)), S(z1)) -> c4(H(h(g(S(z0), S(z1)), f(S(S(S(S(z0)))), 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(z0)), S(z1)))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   H_2, F_2, G_2

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