KILLED
proof of input_6BenuIXuA9.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 373 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 98 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 204 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 460 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 155 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 1665 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 887 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) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms]
    (62) CdtProblem
    (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) 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
   -(s(x), s(y)) -> -(x, y)
   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(x, *(x, y))
   f(s(x)) -> f(-(*(s(s(0)), s(x)), s(s(x))))

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

- => minus
+ => 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:

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

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

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

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

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

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

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

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

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

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

   f_1

(c) The following functions are completely defined:

   times_2
   minus_2
   plus_2

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]

And the following fresh constants:    const

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

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

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

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

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

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

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

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

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

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

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

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

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

   0 => 0
   const => 0

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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(29) 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'

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

   f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z) -{ 5 + 9*z }-> f(s6) :|: s4 >= 0, s4 <= 1 + (1 + 0) + 2 * ((z - 1) * (1 + (1 + 0))), s5 >= 0, s5 <= 1 + (1 + 0) + s4, s6 >= 0, s6 <= s5, z - 1 >= 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s' :|: s' >= 0, s' <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z }-> s'' :|: s'' >= 0, s'' <= z + 0, z >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s3 :|: s1 >= 0, s1 <= z + 2 * ((z' - 2) * z), s2 >= 0, s2 <= z + s1, s3 >= 0, s3 <= z + s2, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(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:

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

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

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

(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:

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

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

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

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

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

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

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

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

Compound Symbols:   c, c1_1, c2, c3_1, c4, c5_2, c6_3


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   +'(0, z0) -> c2
   *'(z0, 0) -> c4
   -'(z0, 0) -> c

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   f(s(z0)) -> f(-(*(s(s(0)), s(z0)), s(s(z0))))
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0)))
K tuples:none
Defined Rule Symbols:   -_2, +_2, *_2, f_1

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3


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

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0)))
K tuples:none
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c6(F(-(*(s(s(0)), s(z0)), s(s(z0)))), -'(*(s(s(0)), s(z0)), s(s(z0))), *'(s(s(0)), s(z0))) by 
   F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1)))
K tuples:none
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z1)) -> c6(F(-(+(s(s(0)), *(s(s(0)), z1)), s(s(z1)))), -'(*(s(s(0)), s(z1)), s(s(z1))), *'(s(s(0)), s(z1))) by 
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c6(-'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
K tuples:none
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3, c6_2


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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
K tuples:none
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3, c_1


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = x_1
   POL(*'(x_1, x_2)) = x_1
   POL(+(x_1, x_2)) = 0
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = [1]
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(F(x_1)) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(s(x_1)) = 0

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3, c_1


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

(63) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(x0)) -> c6(F(-(s(+(s(0), *(s(s(0)), x0))), s(s(x0)))), -'(*(s(s(0)), s(x0)), s(s(x0))), *'(s(s(0)), s(x0))) by F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3, c_1


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

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(*(s(s(0)), s(0)), s(s(0))), *'(s(s(0)), s(0))) by F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c6_3, c_1


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z1))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z1))), s(s(s(z1))))), -'(*(s(s(0)), s(s(z1))), s(s(s(z1)))), *'(s(s(0)), s(s(z1)))) by F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))), *'(s(s(0)), s(z0))) by F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

(73) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(+(s(s(0)), *(s(s(0)), 0)), s(s(0))), *'(s(s(0)), s(0))) by F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
----------------------------------------

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(+(s(s(0)), *(s(s(0)), s(z0))), s(s(s(z0)))), *'(s(s(0)), s(s(z0)))) by F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c(-'(+(s(s(0)), *(s(s(0)), z0)), s(s(z0))))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
S tuples:
   -'(s(z0), s(z1)) -> c1(-'(z0, z1))
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c1_1, c3_1, c5_2, c_1, c6_3


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

(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c1(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
S tuples:
   +'(s(z0), z1) -> c3(+'(z0, z1))
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c3_1, c5_2, c_1, c6_3, c1_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(s(z0), z1) -> c3(+'(z0, z1)) by 
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
S tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1)))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c5_2, c_1, c6_3, c1_1, c3_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c1(-'(s(y0), s(y1))) by 
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), -'(s(+(s(0), *(s(s(0)), 0))), s(s(0))), *'(s(s(0)), s(0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c5_2, c_1, c6_3, c3_1, c1_1


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   +'(s(s(y0)), z1) -> c3(+'(s(y0), z1))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0)))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c5_2, c_1, c6_3, c3_1, c1_1, c6_2


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(s(s(y0)), z1) -> c3(+'(s(y0), z1)) by 
   +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(z0, *(z0, z1))
   +(0, z0) -> z0
   +(s(z0), z1) -> s(+(z0, z1))
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
Tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   F(s(z0)) -> c(-'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0)))
   +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1))
S tuples:
   *'(z0, s(z1)) -> c5(+'(z0, *(z0, z1)), *'(z0, z1))
   F(s(z0)) -> c6(F(-(s(+(s(0), *(s(s(0)), z0))), s(s(z0)))), -'(s(+(s(0), *(s(s(0)), z0))), s(s(z0))), *'(s(s(0)), s(z0)))
   F(s(s(z0))) -> c6(F(-(+(s(s(0)), +(s(s(0)), *(s(s(0)), z0))), s(s(s(z0))))), -'(s(+(s(0), *(s(s(0)), s(z0)))), s(s(s(z0)))), *'(s(s(0)), s(s(z0))))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c1(-'(s(s(y0)), s(s(y1))))
   F(s(0)) -> c6(F(-(+(s(s(0)), 0), s(s(0)))), *'(s(s(0)), s(0)))
   +'(s(s(s(y0))), z1) -> c3(+'(s(s(y0)), z1))
K tuples:
   F(s(x0)) -> c(-'(*(s(s(0)), s(x0)), s(s(x0))))
   F(s(x0)) -> c(*'(s(s(0)), s(x0)))
Defined Rule Symbols:   *_2, +_2, -_2

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

Compound Symbols:   c5_2, c_1, c6_3, c1_1, c6_2, c3_1