KILLED
proof of input_ThNuQyJdtt.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
        (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) CpxTypedWeightedCompleteTrs
        (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) InliningProof [UPPER BOUND(ID), 750 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 166 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 121 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 13 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 152 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 580 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 113 ms]
        (46) CpxRNTS
        (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 3250 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 6592 ms]
        (52) CpxRNTS
    (53) CompletionProof [UPPER BOUND(ID), 0 ms]
    (54) CpxTypedWeightedCompleteTrs
        (55) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
(57) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (66) CdtProblem
    (67) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 165 ms]
    (72) CdtProblem
    (73) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 12 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 20 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 11 ms]
    (130) CdtProblem
    (131) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 33 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 38 ms]
    (156) CdtProblem
    (157) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 62 ms]
    (160) CdtProblem
    (161) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 38 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 22 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 48 ms]
    (168) CdtProblem
    (169) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 63 ms]
    (170) CdtProblem
    (171) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 66 ms]
    (172) CdtProblem
    (173) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) 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(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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(0') -> 0'
   f(s(0')) -> s(0')
   f(s(s(x))) -> p(h(g(x)))
   g(0') -> pair(s(0'), s(0'))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+'(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   f(s(s(x))) -> +'(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+'(p(g(x)), q(g(x))), p(g(x)))

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(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(x))) -> p(h(g(x)))
   g(0) -> pair(s(0), s(0))
   g(s(x)) -> h(g(x))
   h(x) -> pair(+(p(x), q(x)), p(x))
   p(pair(x, y)) -> x
   q(pair(x, y)) -> y
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   f(s(s(x))) -> +(p(g(x)), q(g(x)))
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x)))

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(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(x))) -> p(h(g(x))) [1]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(x)) -> h(g(x)) [1]
   h(x) -> pair(+(p(x), q(x)), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   f(s(s(x))) -> +(p(g(x)), q(g(x))) [1]
   g(s(x)) -> pair(+(p(g(x)), q(g(x))), p(g(x))) [1]

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

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

+ => plus

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

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


The TRS R consists of the following rules:

   f(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(x))) -> p(h(g(x))) [1]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(x)) -> h(g(x)) [1]
   h(x) -> pair(plus(p(x), q(x)), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1]
   g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1]

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

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

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

   f(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(x))) -> p(h(g(x))) [1]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(x)) -> h(g(x)) [1]
   h(x) -> pair(plus(p(x), q(x)), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1]
   g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1]

The TRS has the following type information:
f :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: pair -> 0:s
h :: pair -> pair
g :: 0:s -> pair
pair :: 0:s -> 0:s -> pair
plus :: 0:s -> 0:s -> 0:s
q :: pair -> 0:s

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

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

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

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

   f_1

(c) The following functions are completely defined:

   h_1
   g_1
   p_1
   q_1
   plus_2

Due to the following rules being added:

   p(v0) -> 0 [0]
   q(v0) -> 0 [0]

And the following fresh constants:    const

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

(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(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(x))) -> p(h(g(x))) [1]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(x)) -> h(g(x)) [1]
   h(x) -> pair(plus(p(x), q(x)), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1]
   g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1]
   p(v0) -> 0 [0]
   q(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: pair -> 0:s
h :: pair -> pair
g :: 0:s -> pair
pair :: 0:s -> 0:s -> pair
plus :: 0:s -> 0:s -> 0:s
q :: pair -> 0:s
const :: pair

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

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

(14)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

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

   f(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(0))) -> p(h(pair(s(0), s(0)))) [2]
   f(s(s(s(x')))) -> p(h(h(g(x')))) [2]
   f(s(s(s(x'')))) -> p(h(pair(plus(p(g(x'')), q(g(x''))), p(g(x''))))) [2]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(0)) -> h(pair(s(0), s(0))) [2]
   g(s(s(x1))) -> h(h(g(x1))) [2]
   g(s(s(x2))) -> h(pair(plus(p(g(x2)), q(g(x2))), p(g(x2)))) [2]
   h(pair(x3, y')) -> pair(plus(x3, y'), p(pair(x3, y'))) [3]
   h(pair(x3, y')) -> pair(plus(x3, 0), p(pair(x3, y'))) [2]
   h(pair(x4, y'')) -> pair(plus(0, y''), p(pair(x4, y''))) [2]
   h(x) -> pair(plus(0, 0), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   f(s(s(0))) -> plus(p(pair(s(0), s(0))), q(pair(s(0), s(0)))) [3]
   f(s(s(s(x5)))) -> plus(p(h(g(x5))), q(h(g(x5)))) [3]
   f(s(s(s(x5)))) -> plus(p(h(g(x5))), q(pair(plus(p(g(x5)), q(g(x5))), p(g(x5))))) [3]
   f(s(s(s(x6)))) -> plus(p(pair(plus(p(g(x6)), q(g(x6))), p(g(x6)))), q(h(g(x6)))) [3]
   f(s(s(s(x6)))) -> plus(p(pair(plus(p(g(x6)), q(g(x6))), p(g(x6)))), q(pair(plus(p(g(x6)), q(g(x6))), p(g(x6))))) [3]
   g(s(0)) -> pair(plus(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))) [4]
   g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(h(g(x7)))), p(h(g(x7)))) [4]
   g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(h(g(x7)))), p(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))) [4]
   g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))), p(h(g(x7)))) [4]
   g(s(s(x7))) -> pair(plus(p(h(g(x7))), q(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))), p(pair(plus(p(g(x7)), q(g(x7))), p(g(x7))))) [4]
   g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(h(g(x8)))), p(h(g(x8)))) [4]
   g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(h(g(x8)))), p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))) [4]
   g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))), p(h(g(x8)))) [4]
   g(s(s(x8))) -> pair(plus(p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8)))), q(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))), p(pair(plus(p(g(x8)), q(g(x8))), p(g(x8))))) [4]
   p(v0) -> 0 [0]
   q(v0) -> 0 [0]

The TRS has the following type information:
f :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
p :: pair -> 0:s
h :: pair -> pair
g :: 0:s -> pair
pair :: 0:s -> 0:s -> pair
plus :: 0:s -> 0:s -> 0:s
q :: pair -> 0:s
const :: pair

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

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

   0 => 0
   const => 0

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

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

   f(z) -{ 3 }-> plus(p(h(g(x5))), q(h(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0
   f(z) -{ 3 }-> plus(p(h(g(x5))), q(1 + plus(p(g(x5)), q(g(x5))) + p(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(h(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6))
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6))
   f(z) -{ 3 }-> plus(p(1 + (1 + 0) + (1 + 0)), q(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 2 }-> p(h(h(g(x')))) :|: z = 1 + (1 + (1 + x')), x' >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(x'')), q(g(x''))) + p(g(x'')))) :|: x'' >= 0, z = 1 + (1 + (1 + x''))
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(x2)), q(g(x2))) + p(g(x2))) :|: z = 1 + (1 + x2), x2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + (1 + 0) + (1 + 0)), q(1 + (1 + 0) + (1 + 0))) + p(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + p(1 + x3 + y') :|: y' >= 0, z = 1 + x3 + y', x3 >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + p(1 + x3 + y') :|: y' >= 0, z = 1 + x3 + y', x3 >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + p(1 + x4 + y'') :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + p(x) :|: x >= 0, z = x
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(x5))), q(h(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0
   f(z) -{ 3 }-> plus(p(h(g(x5))), q(1 + plus(p(g(x5)), q(g(x5))) + p(g(x5)))) :|: z = 1 + (1 + (1 + x5)), x5 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(h(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6))
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6))), q(1 + plus(p(g(x6)), q(g(x6))) + p(g(x6)))) :|: x6 >= 0, z = 1 + (1 + (1 + x6))
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(x')))) :|: z = 1 + (1 + (1 + x')), x' >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(x'')), q(g(x''))) + p(g(x'')))) :|: x'' >= 0, z = 1 + (1 + (1 + x''))
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(x1))) :|: z = 1 + (1 + x1), x1 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(x2)), q(g(x2))) + p(g(x2))) :|: z = 1 + (1 + x2), x2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(h(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(h(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(h(g(x7))), q(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7)))) + p(1 + plus(p(g(x7)), q(g(x7))) + p(g(x7))) :|: x7 >= 0, z = 1 + (1 + x7)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(h(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(h(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))), q(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8)))) + p(1 + plus(p(g(x8)), q(g(x8))) + p(g(x8))) :|: x8 >= 0, z = 1 + (1 + x8)
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: x >= 0, z = x, x = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: x >= 0, z = x, v0 >= 0, x = v0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0


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

   { plus }
   { q }
   { p }
   { h }
   { g }
   { f }

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

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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(23) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(24)
Obligation:
Complexity RNTS consisting of the following rules:

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 5 }-> plus(x, y') :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> plus(x, 0) :|: z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 4 }-> plus(0, y) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 3 }-> plus(0, 0) :|: z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 6 }-> 1 + plus(x, y') + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 7 }-> 1 + plus(x, y'') + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + plus(x, 0) + x' :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 5 }-> 1 + plus(x, 0) + 0 :|: z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 5 }-> 1 + plus(0, y) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 6 }-> 1 + plus(0, y') + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + plus(0, 0) + x :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 4 }-> 1 + plus(0, 0) + 0 :|: z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 4 }-> 1 + plus(x3, y') + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + plus(x3, y') + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(x3, 0) + x :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(x3, 0) + 0 :|: y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 3 }-> 1 + plus(0, y'') + x :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 2 }-> 1 + plus(0, y'') + 0 :|: z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 2 }-> 1 + plus(0, 0) + x' :|: z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 1 }-> 1 + plus(0, 0) + 0 :|: z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {h}, {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  q: runtime: O(1) [1], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  h: runtime: ?, size: O(n^1) [1 + 2*z]

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

(45) 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: 4 + z

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + (1 + 0) + (1 + 0))) :|: z = 1 + (1 + 0)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + (1 + 0) + (1 + 0)) :|: z = 1 + 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  q: runtime: O(1) [1], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z]

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0)
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  q: runtime: O(1) [1], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: g
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0)
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  q: runtime: O(1) [1], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z]
  g: runtime: ?, size: EXP

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + y }-> s10 :|: s10 >= 0, s10 <= 0 + y, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   f(z) -{ 10 }-> s21 :|: s20 >= 0, s20 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), s21 >= 0, s21 <= s20, z = 1 + (1 + 0)
   f(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= x + 0, z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   f(z) -{ 6 + y' }-> s8 :|: s8 >= 0, s8 <= x + y', z = 1 + (1 + 0), 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   f(z) -{ 4 }-> s9 :|: s9 >= 0, s9 <= 0 + 0, z = 1 + (1 + 0), v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(h(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 3 }-> plus(p(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3))), q(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(h(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 2 }-> p(h(1 + plus(p(g(z - 3)), q(g(z - 3))) + p(g(z - 3)))) :|: z - 3 >= 0
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 9 }-> s19 :|: s19 >= 0, s19 <= 1 + 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + 0
   g(z) -{ 2 }-> h(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> h(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 7 }-> 1 + s11 + x' :|: s11 >= 0, s11 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0
   g(z) -{ 8 + y'' }-> 1 + s12 + x' :|: s12 >= 0, s12 <= x + y'', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x'' + y'', x'' >= 0, y'' >= 0
   g(z) -{ 6 }-> 1 + s13 + 0 :|: s13 >= 0, s13 <= x + 0, z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s14 + 0 :|: s14 >= 0, s14 <= x + y', z = 1 + 0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 6 }-> 1 + s15 + x :|: s15 >= 0, s15 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0'
   g(z) -{ 7 + y' }-> 1 + s16 + x :|: s16 >= 0, s16 <= 0 + y', z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0, 1 + (1 + 0) + (1 + 0) = 1 + x' + y', x' >= 0, y' >= 0
   g(z) -{ 5 }-> 1 + s17 + 0 :|: s17 >= 0, s17 <= 0 + 0, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', v0'' >= 0, 1 + (1 + 0) + (1 + 0) = v0''
   g(z) -{ 6 + y }-> 1 + s18 + 0 :|: s18 >= 0, s18 <= 0 + y, z = 1 + 0, v0 >= 0, 1 + (1 + 0) + (1 + 0) = v0, v0' >= 0, 1 + (1 + 0) + (1 + 0) = v0', 1 + (1 + 0) + (1 + 0) = 1 + x + y, x >= 0, y >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(h(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(h(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(h(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 4 }-> 1 + plus(p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))), q(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2)))) + p(1 + plus(p(g(z - 2)), q(g(z - 2))) + p(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 5 + y' }-> 1 + s' + x :|: s' >= 0, s' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 4 + y' }-> 1 + s'' + 0 :|: s'' >= 0, s'' <= x3 + y', y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 }-> 1 + s1 + x :|: s1 >= 0, s1 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, 1 + x3 + y' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 }-> 1 + s2 + 0 :|: s2 >= 0, s2 <= x3 + 0, y' >= 0, z = 1 + x3 + y', x3 >= 0, v0 >= 0, 1 + x3 + y' = v0
   h(z) -{ 4 + y'' }-> 1 + s3 + x :|: s3 >= 0, s3 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, 1 + x4 + y'' = 1 + x + y, x >= 0, y >= 0
   h(z) -{ 3 + y'' }-> 1 + s4 + 0 :|: s4 >= 0, s4 <= 0 + y'', z = 1 + x4 + y'', x4 >= 0, y'' >= 0, v0 >= 0, 1 + x4 + y'' = v0
   h(z) -{ 3 }-> 1 + s5 + x' :|: s5 >= 0, s5 <= 0 + 0, z >= 0, z = 1 + x' + y, x' >= 0, y >= 0
   h(z) -{ 2 }-> 1 + s6 + 0 :|: s6 >= 0, s6 <= 0 + 0, z >= 0
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: z >= 0

Function symbols to be analyzed: {g}, {f}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  q: runtime: O(1) [1], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  h: runtime: O(n^1) [4 + z], size: O(n^1) [1 + 2*z]
  g: runtime: INF, size: EXP

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

(53) 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:

   p(v0) -> null_p [0]
   q(v0) -> null_q [0]
   f(v0) -> null_f [0]
   g(v0) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_p, null_q, null_f, null_g, null_plus

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

(54)
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(0) -> 0 [1]
   f(s(0)) -> s(0) [1]
   f(s(s(x))) -> p(h(g(x))) [1]
   g(0) -> pair(s(0), s(0)) [1]
   g(s(x)) -> h(g(x)) [1]
   h(x) -> pair(plus(p(x), q(x)), p(x)) [1]
   p(pair(x, y)) -> x [1]
   q(pair(x, y)) -> y [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   f(s(s(x))) -> plus(p(g(x)), q(g(x))) [1]
   g(s(x)) -> pair(plus(p(g(x)), q(g(x))), p(g(x))) [1]
   p(v0) -> null_p [0]
   q(v0) -> null_q [0]
   f(v0) -> null_f [0]
   g(v0) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
f :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus
0 :: 0:s:null_p:null_q:null_f:null_plus
s :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus
p :: pair:null_g -> 0:s:null_p:null_q:null_f:null_plus
h :: pair:null_g -> pair:null_g
g :: 0:s:null_p:null_q:null_f:null_plus -> pair:null_g
pair :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus -> pair:null_g
plus :: 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus -> 0:s:null_p:null_q:null_f:null_plus
q :: pair:null_g -> 0:s:null_p:null_q:null_f:null_plus
null_p :: 0:s:null_p:null_q:null_f:null_plus
null_q :: 0:s:null_p:null_q:null_f:null_plus
null_f :: 0:s:null_p:null_q:null_f:null_plus
null_g :: pair:null_g
null_plus :: 0:s:null_p:null_q:null_f:null_plus

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

   0 => 0
   null_p => 0
   null_q => 0
   null_f => 0
   null_g => 0
   null_plus => 0

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

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

   f(z) -{ 1 }-> plus(p(g(x)), q(g(x))) :|: x >= 0, z = 1 + (1 + x)
   f(z) -{ 1 }-> p(h(g(x))) :|: x >= 0, z = 1 + (1 + x)
   f(z) -{ 1 }-> 0 :|: z = 0
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   g(z) -{ 1 }-> h(g(x)) :|: x >= 0, z = 1 + x
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + plus(p(g(x)), q(g(x))) + p(g(x)) :|: x >= 0, z = 1 + x
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 0
   h(z) -{ 1 }-> 1 + plus(p(x), q(x)) + p(x) :|: x >= 0, z = x
   p(z) -{ 1 }-> x :|: z = 1 + x + y, x >= 0, y >= 0
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   q(z) -{ 1 }-> y :|: z = 1 + x + y, x >= 0, y >= 0
   q(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   F(0) -> c
   F(s(0)) -> c1
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(0) -> c5
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   H(z0) -> c12(P(z0))
   P(pair(z0, z1)) -> c13
   Q(pair(z0, z1)) -> c14
   +'(z0, 0) -> c15
   +'(z0, s(z1)) -> c16(+'(z0, z1))
S tuples:
   F(0) -> c
   F(s(0)) -> c1
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(0) -> c5
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   H(z0) -> c12(P(z0))
   P(pair(z0, z1)) -> c13
   Q(pair(z0, z1)) -> c14
   +'(z0, 0) -> c15
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   F_1, G_1, H_1, P_1, Q_1, +'_2

Compound Symbols:   c, c1, c2_3, c3_3, c4_3, c5, c6_2, c7_3, c8_3, c9_2, c10_2, c11_2, c12_1, c13, c14, c15, c16_1


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   H(z0) -> c12(P(z0))
   +'(z0, 0) -> c15
   F(s(0)) -> c1
   Q(pair(z0, z1)) -> c14
   F(0) -> c
   P(pair(z0, z1)) -> c13
   G(0) -> c5

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
S tuples:
   F(s(s(z0))) -> c2(P(h(g(z0))), H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), P(g(z0)), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), Q(g(z0)), G(z0))
   G(s(z0)) -> c9(P(g(z0)), G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)), P(z0))
   H(z0) -> c11(+'(p(z0), q(z0)), Q(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   F_1, G_1, H_1, +'_2

Compound Symbols:   c2_3, c3_3, c4_3, c6_2, c7_3, c8_3, c9_2, c10_2, c11_2, c16_1


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

(61) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 8 trailing tuple parts
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   F(s(s(z0))) -> c2(H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
S tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   F(s(s(z0))) -> c2(H(g(z0)), G(z0))
   F(s(s(z0))) -> c3(+'(p(g(z0)), q(g(z0))), G(z0))
   F(s(s(z0))) -> c4(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   G_1, +'_2, F_1, H_1

Compound Symbols:   c6_2, c16_1, c2_2, c3_2, c4_2, c7_2, c8_2, c9_1, c10_1, c11_1


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(G(z0))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(G(z0))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(65) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F(s(s(z0))) -> c(G(z0))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
K tuples:none
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
Defined Rule Symbols:   f_1, g_1, h_1, p_1, q_1, +_2

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(69) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   f(0) -> 0
   f(s(0)) -> s(0)
   f(s(s(z0))) -> p(h(g(z0)))
   f(s(s(z0))) -> +(p(g(z0)), q(g(z0)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(71) 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(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
We considered the (Usable) Rules:none
And the Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1]
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(F(x_1)) = [1] + x_1
   POL(G(x_1)) = x_1
   POL(H(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c16(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1)) = x_1
   POL(g(x_1)) = x_1
   POL(h(x_1)) = [1] + x_1
   POL(p(x_1)) = [1]
   POL(pair(x_1, x_2)) = [1]
   POL(q(x_1)) = [1]
   POL(s(x_1)) = [1] + x_1

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(73) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c7_2, c8_2, c9_1, c10_1, c11_1, c_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0)) by 
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))), G(0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))), G(0))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))), G(0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))), G(0))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c8_2, c9_1, c10_1, c11_1, c_1, c7_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c8_2, c9_1, c10_1, c11_1, c_1, c7_2, c7_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0)) by 
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))), G(0))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))), G(0))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))), G(0))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))), G(0))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c9_1, c10_1, c11_1, c_1, c7_2, c7_1, c8_2


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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c9_1, c10_1, c11_1, c_1, c7_2, c7_1, c8_2, c8_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(z0) -> c10(+'(p(z0), q(z0))) by 
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c11(+'(p(z0), q(z0)))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, H_1, F_1

Compound Symbols:   c6_2, c16_1, c9_1, c11_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(z0) -> c11(+'(p(z0), q(z0))) by 
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, +'_2, F_1, H_1

Compound Symbols:   c6_2, c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(z0)) -> c6(H(g(z0)), G(z0)) by 
   G(s(0)) -> c6(H(pair(s(0), s(0))), G(0))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(0)) -> c6(H(pair(s(0), s(0))), G(0))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2


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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0))) -> c(H(g(z0))) by 
   F(s(s(0))) -> c(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(0))) -> c(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(H(g(z0)))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c6(H(g(z0)), G(z0))
   G(s(z0)) -> c7(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c8(+'(p(g(z0)), q(g(z0))), G(z0))
   G(s(z0)) -> c9(G(z0))
   H(z0) -> c10(+'(p(z0), q(z0)))
   H(z0) -> c11(+'(p(z0), q(z0)))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1


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

(93) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F(s(s(0))) -> c(H(pair(s(0), s(0))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0)))) by 
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) by 
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) by 
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) by 
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) by 
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c7(+'(p(g(0)), q(pair(s(0), s(0))))) by 
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0))))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(g(0)))) by 
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(h(g(z0)))), G(s(z0))) by 
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0))) by 
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(g(s(z0)))), G(s(z0))) by 
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0))) by 
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c8(+'(p(g(0)), q(pair(s(0), s(0))))) by 
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c8_1, c10_1, c11_1, c6_2, c6_1, c_1, c7_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(g(0)))) by 
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c10_1, c11_1, c6_2, c6_1, c_1, c7_1, c8_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(pair(z0, z1)) -> c10(+'(p(pair(z0, z1)), z1)) by 
   H(pair(z0, z1)) -> c10(+'(z0, z1))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1))))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c10_1, c11_1, c6_2, c6_1, c_1, c7_1, c8_1


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

(123) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(pair(z0, z1)) -> c10(+'(z0, q(pair(z0, z1)))) by 
   H(pair(z0, z1)) -> c10(+'(z0, z1))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c11_1, c6_2, c6_1, c_1, c7_1, c8_1, c10_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(pair(z0, z1)) -> c11(+'(p(pair(z0, z1)), z1)) by 
   H(pair(z0, z1)) -> c11(+'(z0, z1))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1))))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, H_1, F_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c11_1, c6_2, c6_1, c_1, c7_1, c8_1, c10_1


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

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace H(pair(z0, z1)) -> c11(+'(z0, q(pair(z0, z1)))) by 
   H(pair(z0, z1)) -> c11(+'(z0, z1))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c6(H(h(g(z0))), G(s(z0))) by 
   G(s(s(x0))) -> c6(H(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), G(s(x0)))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(h(g(z0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(131) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(z0)))) -> c(H(h(g(z0)))) by 
   F(s(s(s(x0)))) -> c(H(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(133) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(0))) -> c(+'(p(g(0)), q(pair(s(0), s(0))))) by 
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(135) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(h(g(z0))))) by 
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))) by 
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(g(0)))) by 
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(141) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(g(s(z0))))) by 
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0))))) by 
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
S tuples:
   +'(z0, s(z1)) -> c16(+'(z0, z1))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   +'_2, G_1, F_1, H_1

Compound Symbols:   c16_1, c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1


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

(145) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(z1)) -> c16(+'(z0, z1)) by 
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c7(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), G(s(z0)))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   F(s(s(z0))) -> c(+'(p(g(z0)), q(g(z0))))
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c9_1, c7_2, c8_2, c6_2, c6_1, c_1, c7_1, c8_1, c10_1, c11_1, c16_1


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

(147) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   G(s(0)) -> c7(+'(p(g(0)), s(0)))
   G(s(0)) -> c8(+'(p(g(0)), s(0)))
   F(s(s(0))) -> c(+'(p(g(0)), s(0)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c9_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(0))) -> c7(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0))) by G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0)))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(z0)) -> c9(G(z0))
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0)))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(z0)) -> c9(G(z0))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c9_1, c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1


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

(151) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace G(s(z0)) -> c9(G(z0)) by 
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c7(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(h(pair(s(0), s(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))), G(s(0)))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(0))) -> c8(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(0))) -> c6(H(h(pair(s(0), s(0)))), G(s(0)))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(0))) -> c7(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))), G(s(0)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1


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

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) by G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1


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

(157) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(0)) -> c7(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0))))) by G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1


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

(161) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(s(0)) -> c7(+'(p(pair(s(0), s(0))), s(0)))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c7(+'(s(0), q(g(0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c7_1, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(0)) -> c7(+'(s(0), q(g(0)))) by G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1, c7_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(pair(z0, z1)) -> c10(+'(z0, z1)) by 
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c10(+'(z0, z1))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c10_1, c11_1, c16_1, c9_1, c1_1, c7_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(pair(z0, z1)) -> c10(+'(z0, z1)) by 
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c11_1, c16_1, c9_1, c1_1, c7_1, c10_1


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

(169) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(pair(z0, z1)) -> c11(+'(z0, z1)) by 
   H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1))))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   H(pair(z0, z1)) -> c11(+'(z0, z1))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))
   H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, H_1, +'_2

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c11_1, c16_1, c9_1, c1_1, c7_1, c10_1


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

(171) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace H(pair(z0, z1)) -> c11(+'(z0, z1)) by 
   H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   G(s(0)) -> c6(H(pair(s(0), s(0))))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))
   H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, +'_2, H_1

Compound Symbols:   c6_2, c6_1, c_1, c7_2, c8_2, c8_1, c16_1, c9_1, c1_1, c7_1, c10_1, c11_1


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

(173) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   G(s(0)) -> c6(H(pair(s(0), s(0))))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), s(0))
   g(s(z0)) -> h(g(z0))
   g(s(z0)) -> pair(+(p(g(z0)), q(g(z0))), p(g(z0)))
   h(z0) -> pair(+(p(z0), q(z0)), p(z0))
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   p(pair(z0, z1)) -> z0
   q(pair(z0, z1)) -> z1
Tuples:
   G(s(s(z0))) -> c6(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), G(s(z0)))
   F(s(s(s(z0)))) -> c(H(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c7(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c7(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c7(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c7(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c7(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(h(g(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(h(g(z0)))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(g(s(x0))), p(g(x0))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))), G(s(s(z0))))
   G(s(s(z0))) -> c8(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(z0))) -> c8(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(z0)))
   G(s(s(x0))) -> c8(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(h(h(g(z0)))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c8(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))), G(s(x0)))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c8(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))), G(s(s(z0))))
   G(s(0)) -> c8(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   G(s(0)) -> c8(+'(s(0), q(g(0))))
   G(s(s(s(z0)))) -> c6(H(h(h(g(z0)))), G(s(s(z0))))
   G(s(s(s(z0)))) -> c6(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), G(s(s(z0))))
   G(s(s(x0))) -> c6(G(s(x0)))
   F(s(s(s(0)))) -> c(H(h(pair(s(0), s(0)))))
   F(s(s(s(s(z0))))) -> c(H(h(h(g(z0)))))
   F(s(s(s(s(z0))))) -> c(H(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(p(pair(s(0), s(0))), q(pair(s(0), s(0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), q(pair(+(p(g(x0)), q(g(x0))), p(g(x0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(h(g(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(h(g(z0)))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(h(g(z0)))))
   F(s(s(s(x0)))) -> c(+'(p(g(s(x0))), p(g(x0))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(s(z0))))) -> c(+'(p(g(s(s(z0)))), q(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0)))))))
   F(s(s(s(0)))) -> c(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   F(s(s(s(z0)))) -> c(+'(p(h(g(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(s(z0)))) -> c(+'(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))))
   F(s(s(0))) -> c(+'(s(0), q(g(0))))
   F(s(s(s(x0)))) -> c(+'(p(pair(+(p(g(x0)), q(g(x0))), p(g(x0)))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(h(g(z0)))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(h(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(x0)))) -> c(+'(+(p(g(x0)), q(g(x0))), q(g(s(x0)))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(h(g(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(g(s(z0)))), p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(h(g(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(g(s(z0))), q(pair(+(p(g(z0)), q(g(z0))), p(g(z0))))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(0)))) -> c(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(h(g(z0))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   F(s(s(s(s(z0))))) -> c(+'(p(pair(+(p(pair(+(p(g(z0)), q(g(z0))), p(g(z0)))), q(g(s(z0)))), p(g(s(z0))))), q(g(s(s(z0))))))
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0)))))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0))))))
   G(s(s(0))) -> c1(+'(p(h(pair(s(0), s(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(g(0))), p(pair(s(0), s(0))))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(g(0)), q(pair(s(0), s(0)))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(pair(+(p(pair(s(0), s(0))), q(g(0))), p(g(0)))), q(g(s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(h(pair(s(0), s(0))))))
   G(s(s(0))) -> c1(H(h(pair(s(0), s(0)))))
   G(s(s(0))) -> c1(+'(p(g(s(0))), q(pair(+(p(pair(s(0), s(0))), q(pair(s(0), s(0)))), p(pair(s(0), s(0)))))))
   G(s(0)) -> c7(+'(s(0), q(pair(s(0), s(0)))))
   H(pair(z0, s(s(y1)))) -> c10(+'(z0, s(s(y1))))
   H(pair(z0, s(s(y1)))) -> c11(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c16(+'(z0, s(y1)))
K tuples:
   G(s(s(y0))) -> c9(G(s(y0)))
   G(s(s(s(y0)))) -> c9(G(s(s(y0))))
   G(s(s(0))) -> c9(G(s(0)))
   G(s(s(s(s(y0))))) -> c9(G(s(s(s(y0)))))
   G(s(s(s(0)))) -> c9(G(s(s(0))))
Defined Rule Symbols:   g_1, h_1, +_2, p_1, q_1

Defined Pair Symbols:   G_1, F_1, +'_2, H_1

Compound Symbols:   c6_2, c_1, c7_2, c8_2, c8_1, c6_1, c16_1, c9_1, c1_1, c7_1, c10_1, c11_1