KILLED
proof of input_D9e6sOjorZ.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 199 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 69 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 315 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 37 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 518 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 125 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 2661 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 1476 ms]
        (50) CpxRNTS
    (51) CompletionProof [UPPER BOUND(ID), 0 ms]
    (52) CpxTypedWeightedCompleteTrs
        (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (54) CpxRNTS
(55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(56) CdtProblem
    (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 155 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 67 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 59 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 8 ms]
    (78) CdtProblem
    (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 76 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms]
    (86) CdtProblem
    (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) 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:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(x, *(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0)))))

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(x, *(x, y))
   twice(0) -> 0
   twice(s(x)) -> s(s(twice(x)))
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   f(s(x)) -> f(-(*(s(s(x)), s(s(x))), +(*(s(x), s(s(x))), s(s(0)))))

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

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

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


The TRS R consists of the following rules:

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

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

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

+ => plus
* => times
- => minus

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

(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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(x, times(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [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:

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

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

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:

   twice_1
   f_1

(c) The following functions are completely defined:

   times_2
   minus_2
   plus_2

Due to the following rules being added:

   minus(v0, v1) -> 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:

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

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

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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(0)) -> plus(x, 0) [2]
   times(x, s(s(y'))) -> plus(x, plus(x, times(x, y'))) [2]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(s(x)) -> f(minus(plus(s(s(x)), times(s(s(x)), s(x))), plus(plus(s(x), times(s(x), s(x))), s(s(0))))) [3]
   minus(v0, v1) -> 0 [0]

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

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 }-> f(minus(plus(1 + (1 + x), times(1 + (1 + x), 1 + x)), plus(plus(1 + x, times(1 + x, 1 + x)), 1 + (1 + 0)))) :|: x >= 0, z = 1 + x
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 2 }-> plus(x, plus(x, times(x, y'))) :|: z' = 1 + (1 + y'), x >= 0, y' >= 0, z = x
   times(z, z') -{ 2 }-> plus(x, 0) :|: x >= 0, z' = 1 + 0, z = x
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x


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

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

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

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


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

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

   { twice }
   { minus }
   { plus }
   { times }
   { f }

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

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

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

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

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(35) 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'

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(37) 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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 2 }-> plus(z, 0) :|: z >= 0, z' = 1 + 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 4 + z + 2*z*z' + 4*z'

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

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

   f(z) -{ 3 }-> f(minus(plus(1 + (1 + (z - 1)), times(1 + (1 + (z - 1)), 1 + (z - 1))), plus(plus(1 + (z - 1), times(1 + (z - 1), 1 + (z - 1))), 1 + (1 + 0)))) :|: z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ 2 }-> plus(z, plus(z, times(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

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

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

   f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= s6, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

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

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= s6, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z']
  f: runtime: ?, size: O(1) [0]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 17 + s8 + s9 + 14*z + 4*z^2 }-> f(s10) :|: s5 >= 0, s5 <= 1 + (1 + (z - 1)) + 2 * ((1 + (z - 1)) * (1 + (1 + (z - 1)))), s6 >= 0, s6 <= 1 + (1 + (z - 1)) + s5, s7 >= 0, s7 <= 1 + (z - 1) + 2 * ((1 + (z - 1)) * (1 + (z - 1))), s8 >= 0, s8 <= 1 + (z - 1) + s7, s9 >= 0, s9 <= s8 + (1 + (1 + 0)), s10 >= 0, s10 <= s6, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s' :|: s' >= 0, s' <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s1 :|: s1 >= 0, s1 <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s4 :|: s2 >= 0, s2 <= z + 2 * ((z' - 2) * z), s3 >= 0, s3 <= z + s2, s4 >= 0, s4 <= z + s3, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 1 + z }-> 1 + (1 + s) :|: s >= 0, s <= 2 * (z - 1), z - 1 >= 0

Function symbols to be analyzed: {f}
Previous analysis results are:
  twice: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z']
  f: runtime: INF, size: O(1) [0]

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

(51) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   f(v0) -> null_f [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   twice(v0) -> null_twice [0]

And the following fresh constants:    null_minus, null_f, null_plus, null_times, null_twice

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

(52)
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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(x, times(x, y)) [1]
   twice(0) -> 0 [1]
   twice(s(x)) -> s(s(twice(x))) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(s(x)) -> f(minus(times(s(s(x)), s(s(x))), plus(times(s(x), s(s(x))), s(s(0))))) [1]
   minus(v0, v1) -> null_minus [0]
   f(v0) -> null_f [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   twice(v0) -> null_twice [0]

The TRS has the following type information:
plus :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice
0 :: 0:s:null_minus:null_plus:null_times:null_twice
s :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice
times :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice
twice :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice
minus :: 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice -> 0:s:null_minus:null_plus:null_times:null_twice
f :: 0:s:null_minus:null_plus:null_times:null_twice -> null_f
null_minus :: 0:s:null_minus:null_plus:null_times:null_twice
null_f :: null_f
null_plus :: 0:s:null_minus:null_plus:null_times:null_twice
null_times :: 0:s:null_minus:null_plus:null_times:null_twice
null_twice :: 0:s:null_minus:null_plus:null_times:null_twice

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

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

   0 => 0
   null_minus => 0
   null_f => 0
   null_plus => 0
   null_times => 0
   null_twice => 0

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

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

   f(z) -{ 1 }-> f(minus(times(1 + (1 + x), 1 + (1 + x)), plus(times(1 + x, 1 + (1 + x)), 1 + (1 + 0)))) :|: x >= 0, z = 1 + x
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 1 }-> plus(x, times(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   times(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   twice(z) -{ 1 }-> 0 :|: z = 0
   twice(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   twice(z) -{ 1 }-> 1 + (1 + twice(x)) :|: x >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))))
Tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(0) -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   +'(0, z0) -> c
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(0) -> c4
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(z0, 0) -> c6
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, twice_1, -_2, f_1

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5_1, c6, c7_1, c8_3, c9_4


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

(57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   *'(z0, 0) -> c2
   -'(z0, 0) -> c6
   TWICE(0) -> c4
   +'(0, z0) -> c

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))))
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, twice_1, -_2, f_1

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4


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

(59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   twice(0) -> 0
   twice(s(z0)) -> s(s(twice(z0)))
   f(s(z0)) -> f(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TWICE(s(z0)) -> c5(TWICE(z0))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = 0
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = 0
   POL(TWICE(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(s(x_1)) = [1] + x_1

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by 
   F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by 
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0)) -> c8(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by 
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4, c8_1


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = x_1
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = [1]
   POL(TWICE(x_1)) = 0
   POL(c1(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(s(x_1)) = 0

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c9_4, c8_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by 
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c8_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = x_1
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = [1]
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(s(x_1)) = 0

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0)) -> c9(F(-(*(s(s(x0)), s(s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by 
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c9_3


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c2_1


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = x_1
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = [1]
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(s(x_1)) = 0

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c9_4, c8_3, c8_1, c_1, c2_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by 
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = x_1
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = [1]
   POL(TWICE(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_3 + x_4
   POL(s(x_1)) = 0

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1


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

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1


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

(89) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z1)) -> c8(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c8_1, c_1, c9_4, c2_1, c9_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c8(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
S tuples:
   +'(s(z0), z1) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   +'_2, *'_2, TWICE_1, -'_2, F_1

Compound Symbols:   c1_1, c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1


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

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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1, c1_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c8(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), *'(s(s(x0)), s(s(x0)))) by F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c8_3, c_1, c9_4, c2_1, c9_1, c8_1, c1_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z1)) -> c8(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), *'(s(s(z1)), s(s(z1)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(*(s(z0), s(s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z1)) -> c9(F(-(*(s(s(z1)), s(s(z1))), +(+(s(z1), +(s(z1), *(s(z1), z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0)))) by F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c9_4, c2_1, c9_1, c8_3, c8_1, c1_1


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c9(F(-(+(s(s(x0)), *(s(s(x0)), s(x0))), +(+(s(x0), *(s(x0), s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   TWICE(s(z0)) -> c5(TWICE(z0))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   TWICE(s(z0)) -> c5(TWICE(z0))
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, TWICE_1, -'_2, F_1, +'_2

Compound Symbols:   c3_2, c5_1, c7_1, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TWICE(s(z0)) -> c5(TWICE(z0)) by 
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(z0), s(z1)) -> c7(-'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, -'_2, F_1, +'_2, TWICE_1

Compound Symbols:   c3_2, c7_1, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c7(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1, c7_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c9(F(-(s(+(s(x0), *(s(s(x0)), s(x0)))), +(*(s(x0), s(s(x0))), s(s(0))))), -'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))), +'(*(s(x0), s(s(x0))), s(s(0))), *'(s(x0), s(s(x0)))) by F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1, c7_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z1)) -> c9(F(-(+(s(s(z1)), +(s(s(z1)), *(s(s(z1)), z1))), +(*(s(z1), s(s(z1))), s(s(0))))), -'(*(s(s(z1)), s(s(z1))), +(*(s(z1), s(s(z1))), s(s(0)))), +'(*(s(z1), s(s(z1))), s(s(0))), *'(s(z1), s(s(z1)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0))))) by F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c7_1, c9_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c1_1, c9_4, c5_1, c7_1, c9_1, c8_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   +'(s(s(y0)), z1) -> c1(+'(s(y0), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c1_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1


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

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

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1, c1_1


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c9_4, c5_1, c7_1, c9_1, c8_1, c1_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))), *'(s(s(z0)), s(s(z0)))) by F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c2(+'(+(s(z0), *(s(z0), s(z0))), s(s(0)))) by F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(y0))) -> c5(TWICE(s(y0)))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, TWICE_1, -'_2, +'_2

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c5_1, c7_1, c9_1, c8_3, c8_1, c1_1


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

(155) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace TWICE(s(s(y0))) -> c5(TWICE(s(y0))) by 
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, -'_2, +'_2, TWICE_1

Compound Symbols:   c3_2, c_1, c2_1, c9_4, c7_1, c9_1, c8_3, c8_1, c1_1, c5_1


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, -'_2, +'_2, TWICE_1

Compound Symbols:   c3_2, c_1, c2_1, c7_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, -'_2, +'_2, TWICE_1

Compound Symbols:   c3_2, c_1, c2_1, c7_1, c9_4, c9_1, c8_3, c8_1, c1_1, c5_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(+(s(z0), *(s(z0), s(z0))), s(s(0))), *'(s(z0), s(s(z0)))) by F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0)))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, -'_2, +'_2, TWICE_1

Compound Symbols:   c3_2, c_1, c2_1, c7_1, c9_1, c8_3, c8_1, c1_1, c9_4, c5_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))) by F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1)))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, -'_2, +'_2, TWICE_1

Compound Symbols:   c3_2, c_1, c2_1, c7_1, c8_3, c8_1, c1_1, c9_4, c5_1, c9_1


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

(165) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c7(-'(s(y0), s(y1))) by 
   -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
Tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c2(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   F(s(z0)) -> c2(+'(s(+(z0, *(s(z0), s(z0)))), s(s(0))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(-'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   +'(s(s(s(y0))), z1) -> c1(+'(s(s(y0)), z1))
   F(s(z0)) -> c8(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c8(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0)))), *'(s(s(z0)), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(s(+(z0, *(s(z0), s(z0)))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(*(s(s(z0)), s(s(z0))), +(+(s(z0), +(s(z0), *(s(z0), z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), *(s(s(z0)), s(z0))), +(+(s(z0), *(s(z0), s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(s(+(s(z0), *(s(s(z0)), s(z0)))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   F(s(z0)) -> c9(F(-(+(s(s(z0)), +(s(s(z0)), *(s(s(z0)), z0))), +(*(s(z0), s(s(z0))), s(s(0))))), -'(*(s(s(z0)), s(s(z0))), +(*(s(z0), s(s(z0))), s(s(0)))), +'(s(+(z0, *(s(z0), s(z0)))), s(s(0))), *'(s(z0), s(s(z0))))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c7(-'(s(s(y0)), s(s(y1))))
K tuples:
   F(s(x0)) -> c8(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c(*'(s(s(x0)), s(s(x0))))
   F(s(x0)) -> c2(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   F(s(x0)) -> c2(+'(*(s(x0), s(s(x0))), s(s(0))))
   F(s(x0)) -> c2(*'(s(x0), s(s(x0))))
   F(s(x0)) -> c9(-'(*(s(s(x0)), s(s(x0))), +(*(s(x0), s(s(x0))), s(s(0)))))
   TWICE(s(s(s(y0)))) -> c5(TWICE(s(s(y0))))
Defined Rule Symbols:   *_2, +_2, -_2

Defined Pair Symbols:   *'_2, F_1, +'_2, TWICE_1, -'_2

Compound Symbols:   c3_2, c_1, c2_1, c8_3, c8_1, c1_1, c9_4, c5_1, c9_1, c7_1