KILLED
proof of input_ql8qjBHEJU.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 428 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 143 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 967 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 250 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 264 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 796 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 16 ms]
        (44) CpxRNTS
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 136 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (58) CdtProblem
    (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 19 ms]
    (126) CdtProblem
    (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 4 ms]
    (128) CdtProblem
    (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) 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:

   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))

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:

   +'(x, 0') -> x
   +'(0', x) -> x
   +'(s(x), s(y)) -> s(s(+'(x, y)))
   *'(x, 0') -> 0'
   *'(0', x) -> 0'
   *'(s(x), s(y)) -> s(+'(*'(x, y), +'(x, y)))
   sum(nil) -> 0'
   sum(cons(x, l)) -> +'(x, sum(l))
   prod(nil) -> s(0')
   prod(cons(x, l)) -> *'(x, prod(l))

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:

   +(x, 0) -> x
   +(0, x) -> x
   +(s(x), s(y)) -> s(s(+(x, y)))
   *(x, 0) -> 0
   *(0, x) -> 0
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y)))
   sum(nil) -> 0
   sum(cons(x, l)) -> +(x, sum(l))
   prod(nil) -> s(0)
   prod(cons(x, l)) -> *(x, prod(l))

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:

   +(x, 0) -> x [1]
   +(0, x) -> x [1]
   +(s(x), s(y)) -> s(s(+(x, y))) [1]
   *(x, 0) -> 0 [1]
   *(0, x) -> 0 [1]
   *(s(x), s(y)) -> s(+(*(x, y), +(x, y))) [1]
   sum(nil) -> 0 [1]
   sum(cons(x, l)) -> +(x, sum(l)) [1]
   prod(nil) -> s(0) [1]
   prod(cons(x, l)) -> *(x, prod(l)) [1]

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

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

+ => plus
* => times

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

(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(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(s(x), s(y)) -> s(s(plus(x, y))) [1]
   times(x, 0) -> 0 [1]
   times(0, x) -> 0 [1]
   times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1]
   sum(nil) -> 0 [1]
   sum(cons(x, l)) -> plus(x, sum(l)) [1]
   prod(nil) -> s(0) [1]
   prod(cons(x, l)) -> times(x, prod(l)) [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(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(s(x), s(y)) -> s(s(plus(x, y))) [1]
   times(x, 0) -> 0 [1]
   times(0, x) -> 0 [1]
   times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1]
   sum(nil) -> 0 [1]
   sum(cons(x, l)) -> plus(x, sum(l)) [1]
   prod(nil) -> s(0) [1]
   prod(cons(x, l)) -> times(x, prod(l)) [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
sum :: nil:cons -> 0:s
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
prod :: nil:cons -> 0:s

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

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

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

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

(c) The following functions are completely defined:

   times_2
   plus_2
   prod_1
   sum_1

Due to the following rules being added:
none

And the following fresh constants: none

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

(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(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(s(x), s(y)) -> s(s(plus(x, y))) [1]
   times(x, 0) -> 0 [1]
   times(0, x) -> 0 [1]
   times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1]
   sum(nil) -> 0 [1]
   sum(cons(x, l)) -> plus(x, sum(l)) [1]
   prod(nil) -> s(0) [1]
   prod(cons(x, l)) -> times(x, prod(l)) [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
sum :: nil:cons -> 0:s
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
prod :: nil:cons -> 0:s

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(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(s(x), s(y)) -> s(s(plus(x, y))) [1]
   times(x, 0) -> 0 [1]
   times(0, x) -> 0 [1]
   times(s(x), s(0)) -> s(plus(0, x)) [3]
   times(s(0), s(0)) -> s(plus(0, 0)) [3]
   times(s(0), s(0)) -> s(plus(0, 0)) [3]
   times(s(0), s(y)) -> s(plus(0, y)) [3]
   times(s(s(x')), s(s(y'))) -> s(plus(s(plus(times(x', y'), plus(x', y'))), s(s(plus(x', y'))))) [3]
   sum(nil) -> 0 [1]
   sum(cons(x, nil)) -> plus(x, 0) [2]
   sum(cons(x, cons(x'', l'))) -> plus(x, plus(x'', sum(l'))) [2]
   prod(nil) -> s(0) [1]
   prod(cons(x, nil)) -> times(x, s(0)) [2]
   prod(cons(x, cons(x1, l''))) -> times(x, times(x1, prod(l''))) [2]

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
sum :: nil:cons -> 0:s
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
prod :: nil:cons -> 0:s

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
   nil => 0

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

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

   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(x, 1 + 0) :|: x >= 0, z = 1 + x + 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(x, 0) :|: x >= 0, z = 1 + x + 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, x) :|: x >= 0, z' = 1 + 0, z = 1 + x
   times(z, z') -{ 3 }-> 1 + plus(0, y) :|: z' = 1 + y, z = 1 + 0, y >= 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(x', y'), plus(x', y')), 1 + (1 + plus(x', y'))) :|: z' = 1 + (1 + y'), x' >= 0, y' >= 0, z = 1 + (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:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0


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

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

   { plus }
   { times }
   { sum }
   { prod }

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {plus}, {times}, {sum}, {prod}

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

(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:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {plus}, {times}, {sum}, {prod}

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

(23) 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'

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(z - 1, z' - 1)) :|: z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 2 }-> plus(z - 1, 0) :|: z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 3 }-> 1 + plus(0, 0) :|: z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z - 1) :|: z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 3 }-> 1 + plus(0, z' - 1) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 }-> 1 + plus(1 + plus(times(z - 2, z' - 2), plus(z - 2, z' - 2)), 1 + (1 + plus(z - 2, z' - 2))) :|: z - 2 >= 0, z' - 2 >= 0

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

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

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

(29) 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: 3 + 5*z + 2*z*z' + 2*z^2 + z'

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {times}, {sum}, {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: ?, size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']

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

(31) 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: 15 + 2*z + 4*z*z' + 2*z^2 + z'

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 2 }-> times(z - 1, 1 + 0) :|: z - 1 >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + 2*z' }-> 1 + plus(1 + plus(times(z - 2, z' - 2), s2), 1 + (1 + s3)) :|: s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {sum}, {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + 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:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {sum}, {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {sum}, {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']
  sum: runtime: ?, size: O(n^1) [z]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: sum
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z + 2*z^2

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 2 }-> plus(x, plus(x'', sum(l'))) :|: x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']
  sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [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:

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']
  sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']
  sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z]
  prod: runtime: ?, size: INF

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 2 + z' }-> 1 + (1 + s) :|: s >= 0, s <= z - 1 + (z' - 1), z - 1 >= 0, z' - 1 >= 0
   prod(z) -{ 14 + 2*z + 2*z^2 }-> s8 :|: s8 >= 0, s8 <= 5 * (z - 1) + (1 + 0) + 2 * ((1 + 0) * (z - 1)) + 2 * ((z - 1) * (z - 1)) + 3, z - 1 >= 0
   prod(z) -{ 2 }-> times(x, times(x1, prod(l''))) :|: z = 1 + x + (1 + x1 + l''), x1 >= 0, l'' >= 0, x >= 0
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 12 + 4*l' + 2*l'^2 + s10 + s9 }-> s11 :|: s9 >= 0, s9 <= l', s10 >= 0, s10 <= x'' + s9, s11 >= 0, s11 <= x + s10, x >= 0, l' >= 0, x'' >= 0, z = 1 + x + (1 + x'' + l')
   sum(z) -{ 4 }-> s4 :|: s4 >= 0, s4 <= z - 1 + 0, z - 1 >= 0
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 4 + z }-> 1 + s' :|: s' >= 0, s' <= 0 + (z - 1), z - 1 >= 0, z' = 1 + 0
   times(z, z') -{ 5 }-> 1 + s'' :|: s'' >= 0, s'' <= 0 + 0, z = 1 + 0, z' = 1 + 0
   times(z, z') -{ 4 + z' }-> 1 + s1 :|: s1 >= 0, s1 <= 0 + (z' - 1), z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 42 + s2 + s3 + -14*z + 4*z*z' + 2*z^2 + -5*z' }-> 1 + s7 :|: s5 >= 0, s5 <= 5 * (z - 2) + (z' - 2) + 2 * ((z' - 2) * (z - 2)) + 2 * ((z - 2) * (z - 2)) + 3, s6 >= 0, s6 <= s5 + s2, s7 >= 0, s7 <= 1 + s6 + (1 + (1 + s3)), s2 >= 0, s2 <= z - 2 + (z' - 2), s3 >= 0, s3 <= z - 2 + (z' - 2), z - 2 >= 0, z' - 2 >= 0

Function symbols to be analyzed: {prod}
Previous analysis results are:
  plus: runtime: O(n^1) [2 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [15 + 2*z + 4*z*z' + 2*z^2 + z'], size: O(n^2) [3 + 5*z + 2*z*z' + 2*z^2 + z']
  sum: runtime: O(n^2) [6 + 4*z + 2*z^2], size: O(n^1) [z]
  prod: runtime: INF, size: INF

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

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

And the following fresh constants: none

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

(46)
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(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(s(x), s(y)) -> s(s(plus(x, y))) [1]
   times(x, 0) -> 0 [1]
   times(0, x) -> 0 [1]
   times(s(x), s(y)) -> s(plus(times(x, y), plus(x, y))) [1]
   sum(nil) -> 0 [1]
   sum(cons(x, l)) -> plus(x, sum(l)) [1]
   prod(nil) -> s(0) [1]
   prod(cons(x, l)) -> times(x, prod(l)) [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
sum :: nil:cons -> 0:s
nil :: nil:cons
cons :: 0:s -> nil:cons -> nil:cons
prod :: nil:cons -> 0:s

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

(47) 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
   nil => 0

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

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

   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + (1 + plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   prod(z) -{ 1 }-> times(x, prod(l)) :|: x >= 0, l >= 0, z = 1 + x + l
   prod(z) -{ 1 }-> 1 + 0 :|: z = 0
   sum(z) -{ 1 }-> plus(x, sum(l)) :|: x >= 0, l >= 0, z = 1 + x + l
   sum(z) -{ 1 }-> 0 :|: z = 0
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   times(z, z') -{ 1 }-> 0 :|: z' = x, x >= 0, z = 0
   times(z, z') -{ 1 }-> 1 + plus(times(x, y), plus(x, y)) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x

Only complete derivations are relevant for the runtime complexity.

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(z0, 0) -> c
   +'(0, z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0) -> c3
   *'(0, z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
S tuples:
   +'(z0, 0) -> c
   +'(0, z0) -> c1
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(z0, 0) -> c3
   *'(0, z0) -> c4
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(nil) -> c7
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(nil) -> c9
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
K tuples:none
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c, c1, c2_1, c3, c4, c5_2, c6_2, c7, c8_2, c9, c10_2


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   SUM(nil) -> c7
   +'(0, z0) -> c1
   *'(z0, 0) -> c3
   +'(z0, 0) -> c
   *'(0, z0) -> c4
   PROD(nil) -> c9

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   *'(s(z0), s(z1)) -> c6(+'(*(z0, z1), +(z0, z1)), +'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
K tuples:none
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c5_2, c6_2, c8_2, c10_2


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c5_2, c8_2, c10_2, c_1


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

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1 + x_2
   POL(*'(x_1, x_2)) = [1]
   POL(+(x_1, x_2)) = [1] + x_1
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(PROD(x_1)) = x_1
   POL(SUM(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(prod(x_1)) = [1] + x_1
   POL(s(x_1)) = [1]
   POL(sum(x_1)) = [1] + x_1

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c5_2, c8_2, c10_2, c_1


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1 + x_2
   POL(*'(x_1, x_2)) = 0
   POL(+(x_1, x_2)) = [1] + x_1
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(PROD(x_1)) = x_1
   POL(SUM(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(prod(x_1)) = [1] + x_1
   POL(s(x_1)) = [1]
   POL(sum(x_1)) = [1] + x_1

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c5_2, c8_2, c10_2, c_1


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

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1 + x_2
   POL(*'(x_1, x_2)) = [1] + x_1
   POL(+(x_1, x_2)) = x_1 + x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(PROD(x_1)) = x_1
   POL(SUM(x_1)) = x_1
   POL(c(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(prod(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1
   POL(sum(x_1)) = [1] + x_1

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c5_2, c8_2, c10_2, c_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1)) by 
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0))
   *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0))
   *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c5(+'(*(z0, z1), +(z0, z1)), *'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, SUM_1, PROD_1, *'_2

Compound Symbols:   c2_1, c8_2, c10_2, c_1, c5_2


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

(63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(0), s(z0)) -> c5(+'(0, +(0, z0)), *'(0, z0))
   *'(s(z0), s(0)) -> c5(+'(0, +(z0, 0)), *'(z0, 0))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0), *'(z0, 0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0), *'(0, z0))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, SUM_1, PROD_1, *'_2

Compound Symbols:   c2_1, c8_2, c10_2, c_1, c5_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, SUM_1, PROD_1, *'_2

Compound Symbols:   c2_1, c8_2, c10_2, c_1, c5_2, c5_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1)) by 
   SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   SUM(cons(z0, z1)) -> c8(+'(z0, sum(z1)), SUM(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, PROD_1, *'_2, SUM_1

Compound Symbols:   c2_1, c10_2, c_1, c5_2, c5_1, c8_2


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUM(cons(x0, nil)) -> c8(+'(x0, 0), SUM(nil))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, PROD_1, *'_2, SUM_1

Compound Symbols:   c2_1, c10_2, c_1, c5_2, c5_1, c8_2


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1)) by 
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)), PROD(nil))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)), PROD(nil))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1))) by 
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(z0), s(0)) -> c(+'(0, +(z0, 0)))
   *'(s(0), s(z0)) -> c(+'(0, +(0, z0)))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(z0), s(0)) -> c(+'(0, +(z0, 0)))
   *'(s(0), s(z0)) -> c(+'(0, +(0, z0)))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(*(z0, z1), +(z0, z1)))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(z0, z1)) -> c10(*'(z0, prod(z1)), PROD(z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1


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

(77) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(z0), s(0)) -> c(+'(0, +(z0, 0)))
   *'(s(0), s(z0)) -> c(+'(0, +(0, z0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c5(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))), *'(s(z0), s(z1))) by 
   *'(s(s(z0)), s(s(0))) -> c5(+'(*(s(z0), s(0)), s(s(z0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(*(s(0), s(z0)), s(s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(z0)), s(s(0))) -> c5(+'(*(s(z0), s(0)), s(s(z0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(*(s(0), s(z0)), s(s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1


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

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1)))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_2, c5_1, c8_2, c10_2, c10_1, c1_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))), *'(s(z0), s(z1))) by 
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(0, +(0, z0))), +(s(0), s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c5(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))), *'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c5(+'(s(+(0, +(0, z0))), +(s(0), s(z0))), *'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(z0), s(0)) -> c5(+'(*(z0, 0), z0)) by 
   *'(s(z0), s(0)) -> c5(+'(0, z0))
   *'(s(0), s(0)) -> c5(+'(0, 0))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(z0), s(0)) -> c5(+'(0, z0))
   *'(s(0), s(0)) -> c5(+'(0, 0))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(0), s(0)) -> c5(+'(0, 0))
   *'(s(z0), s(0)) -> c5(+'(0, z0))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c5_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(0), s(z0)) -> c5(+'(*(0, z0), z0)) by 
   *'(s(0), s(0)) -> c5(+'(0, 0))
   *'(s(0), s(z0)) -> c5(+'(0, z0))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(0), s(0)) -> c5(+'(0, 0))
   *'(s(0), s(z0)) -> c5(+'(0, z0))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1, c5_1


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

(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(0), s(z0)) -> c5(+'(0, z0))
   *'(s(0), s(0)) -> c5(+'(0, 0))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1)))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, SUM_1, PROD_1

Compound Symbols:   c2_1, c_1, c8_2, c10_2, c10_1, c5_2, c1_1, c3_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM(cons(x0, cons(z0, z1))) -> c8(+'(x0, +(z0, sum(z1))), SUM(cons(z0, z1))) by 
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)), SUM(cons(x1, nil)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)), SUM(cons(x1, nil)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_2, c10_1, c5_2, c1_1, c3_1, c8_2


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1)))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_2, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PROD(cons(x0, cons(z0, z1))) -> c10(*'(x0, *(z0, prod(z1))), PROD(cons(z0, z1))) by 
   PROD(cons(x0, cons(0, x2))) -> c10(*'(x0, 0), PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(0, x2))) -> c10(*'(x0, 0), PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, nil))) -> c10(*'(x0, *(x1, s(0))), PROD(cons(x1, nil)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2


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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(z0), s(0)) -> c(+'(*(z0, 0), z0)) by 
   *'(s(z0), s(0)) -> c(+'(0, z0))
   *'(s(0), s(0)) -> c(+'(0, 0))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(z0), s(0)) -> c(+'(0, z0))
   *'(s(0), s(0)) -> c(+'(0, 0))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(z0), s(0)) -> c(+'(0, z0))
   *'(s(0), s(0)) -> c(+'(0, 0))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(0), s(z0)) -> c(+'(*(0, z0), z0))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(0), s(z0)) -> c(+'(*(0, z0), z0)) by 
   *'(s(0), s(0)) -> c(+'(0, 0))
   *'(s(0), s(z0)) -> c(+'(0, z0))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(0), s(0)) -> c(+'(0, 0))
   *'(s(0), s(z0)) -> c(+'(0, z0))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(111) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   *'(s(0), s(z0)) -> c(+'(0, z0))
   *'(s(0), s(0)) -> c(+'(0, 0))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c(+'(*(s(z0), s(z1)), s(s(+(z0, z1))))) by 
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))) by 
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
S tuples:
   +'(s(z0), s(z1)) -> c2(+'(z0, z1))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   +'_2, *'_2, PROD_1, SUM_1

Compound Symbols:   c2_1, c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(s(z0), s(z1)) -> c2(+'(z0, z1)) by 
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
S tuples:
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
K tuples:
   *'(s(z0), s(z1)) -> c(+'(z0, z1))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, PROD_1, SUM_1, +'_2

Compound Symbols:   c_1, c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c2_1


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

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

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
S tuples:
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
K tuples:
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   PROD_1, *'_2, SUM_1, +'_2

Compound Symbols:   c10_1, c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c2_1


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

(121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   PROD(cons(x0, nil)) -> c10(*'(x0, s(0)))
   *'(s(s(0)), s(s(z0))) -> c3(*'(s(0), s(z0)))
   *'(s(s(z0)), s(s(0))) -> c3(*'(s(z0), s(0)))
   *'(s(s(0)), s(s(z0))) -> c1(*'(s(0), s(z0)))
   PROD(cons(x0, cons(x1, nil))) -> c4(PROD(cons(x1, nil)))
   *'(s(s(z0)), s(s(0))) -> c1(*'(s(z0), s(0)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2)))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
S tuples:
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
K tuples:
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, SUM_1, PROD_1, +'_2

Compound Symbols:   c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c10_1, c4_1, c_1, c2_1


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

(123) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PROD(cons(x0, cons(0, x2))) -> c10(PROD(cons(0, x2))) by 
   PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1))))
   PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil))))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
   PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1))))
   PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil))))
S tuples:
   +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1)))
K tuples:
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, SUM_1, PROD_1, +'_2

Compound Symbols:   c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c2_1, c10_1


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

(125) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(s(s(y0)), s(s(y1))) -> c2(+'(s(y0), s(y1))) by 
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
   PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1))))
   PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil))))
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
S tuples:
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
K tuples:
   *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, SUM_1, PROD_1, +'_2

Compound Symbols:   c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1


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

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace *'(s(s(s(y0))), s(s(s(y1)))) -> c(+'(s(s(y0)), s(s(y1)))) by 
   *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1)))))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1))))
   PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil))))
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
   *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
K tuples:
   *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, SUM_1, PROD_1, +'_2

Compound Symbols:   c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1


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

(129) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PROD(cons(z0, cons(0, cons(0, y1)))) -> c10(PROD(cons(0, cons(0, y1)))) by 
   PROD(cons(z0, cons(0, cons(0, cons(y2, y3))))) -> c10(PROD(cons(0, cons(0, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, nil)))) -> c10(PROD(cons(0, cons(0, nil))))
   PROD(cons(z0, cons(0, cons(0, cons(y1, cons(y2, y3)))))) -> c10(PROD(cons(0, cons(0, cons(y1, cons(y2, y3))))))
   PROD(cons(z0, cons(0, cons(0, cons(0, y1))))) -> c10(PROD(cons(0, cons(0, cons(0, y1)))))
   PROD(cons(z0, cons(0, cons(0, cons(y1, nil))))) -> c10(PROD(cons(0, cons(0, cons(y1, nil)))))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(0, z0) -> z0
   +(s(z0), s(z1)) -> s(s(+(z0, z1)))
   *(z0, 0) -> 0
   *(0, z0) -> 0
   *(s(z0), s(z1)) -> s(+(*(z0, z1), +(z0, z1)))
   sum(nil) -> 0
   sum(cons(z0, z1)) -> +(z0, sum(z1))
   prod(nil) -> s(0)
   prod(cons(z0, z1)) -> *(z0, prod(z1))
Tuples:
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(z1))) -> c5(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))), *'(s(z0), s(z1)))
   *'(s(s(z0)), s(s(0))) -> c1(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c1(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c5(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))), *'(s(s(z0)), s(s(z1))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(z0)), s(s(0))) -> c3(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c3(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   SUM(cons(x0, cons(0, x2))) -> c8(+'(x0, sum(x2)), SUM(cons(0, x2)))
   SUM(cons(x0, cons(x1, cons(z0, z1)))) -> c8(+'(x0, +(x1, +(z0, sum(z1)))), SUM(cons(x1, cons(z0, z1))))
   SUM(cons(x0, cons(x1, nil))) -> c8(+'(x0, +(x1, 0)))
   PROD(cons(x0, cons(x1, cons(z0, z1)))) -> c10(*'(x0, *(x1, *(z0, prod(z1)))), PROD(cons(x1, cons(z0, z1))))
   PROD(cons(x0, cons(x1, nil))) -> c4(*'(x0, *(x1, s(0))))
   *'(s(s(z0)), s(s(0))) -> c(+'(*(s(z0), s(0)), s(s(z0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(*(s(0), s(z0)), s(s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(*(s(s(z0)), s(s(z1))), s(s(s(s(+(z0, z1)))))))
   *'(s(s(z0)), s(s(z1))) -> c(+'(s(+(*(z0, z1), +(z0, z1))), s(s(+(z0, z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(*(z0, 0), z0)), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(*(0, z0), z0)), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(*(s(z0), s(z1)), s(s(+(z0, z1))))), +(s(s(z0)), s(s(z1)))))
   *'(s(s(z0)), s(s(0))) -> c(+'(s(+(0, +(z0, 0))), +(s(z0), s(0))))
   *'(s(s(0)), s(s(z0))) -> c(+'(s(+(0, +(0, z0))), +(s(0), s(z0))))
   *'(s(s(s(z0))), s(s(s(z1)))) -> c(+'(s(+(s(+(*(z0, z1), +(z0, z1))), +(s(z0), s(z1)))), +(s(s(z0)), s(s(z1)))))
   PROD(cons(z0, cons(0, cons(y1, cons(y2, y3))))) -> c10(PROD(cons(0, cons(y1, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(y1, nil)))) -> c10(PROD(cons(0, cons(y1, nil))))
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
   *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1)))))
   PROD(cons(z0, cons(0, cons(0, cons(y2, y3))))) -> c10(PROD(cons(0, cons(0, cons(y2, y3)))))
   PROD(cons(z0, cons(0, cons(0, nil)))) -> c10(PROD(cons(0, cons(0, nil))))
   PROD(cons(z0, cons(0, cons(0, cons(y1, cons(y2, y3)))))) -> c10(PROD(cons(0, cons(0, cons(y1, cons(y2, y3))))))
   PROD(cons(z0, cons(0, cons(0, cons(0, y1))))) -> c10(PROD(cons(0, cons(0, cons(0, y1)))))
   PROD(cons(z0, cons(0, cons(0, cons(y1, nil))))) -> c10(PROD(cons(0, cons(0, cons(y1, nil)))))
S tuples:
   +'(s(s(s(y0))), s(s(s(y1)))) -> c2(+'(s(s(y0)), s(s(y1))))
K tuples:
   *'(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c(+'(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   +_2, *_2, sum_1, prod_1

Defined Pair Symbols:   *'_2, SUM_1, PROD_1, +'_2

Compound Symbols:   c5_2, c1_1, c3_1, c8_2, c8_1, c10_2, c4_1, c_1, c10_1, c2_1