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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 203 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 91 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 77 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 524 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 170 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 1124 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 116 ms]
        (42) CpxRNTS
    (43) CompletionProof [UPPER BOUND(ID), 0 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
(47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(48) CdtProblem
    (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 725 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 558 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtLeafRemovalProof [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) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms]
    (88) CdtProblem
    (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) 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:

   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   times(0, y) -> 0
   times(x, 0) -> 0
   times(s(x), y) -> plus(times(x, y), y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   fac(s(x)) -> times(fac(p(s(x))), 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:

   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   times(0', y) -> 0'
   times(x, 0') -> 0'
   times(s(x), y) -> plus(times(x, y), y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0')) -> 0'
   fac(s(x)) -> times(fac(p(s(x))), 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:

   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   times(0, y) -> 0
   times(x, 0) -> 0
   times(s(x), y) -> plus(times(x, y), y)
   p(s(s(x))) -> s(p(s(x)))
   p(s(0)) -> 0
   fac(s(x)) -> times(fac(p(s(x))), 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:

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(s(0)) -> 0 [1]
   fac(s(x)) -> times(fac(p(s(x))), s(x)) [1]

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

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

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

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(s(0)) -> 0 [1]
   fac(s(x)) -> times(fac(p(s(x))), s(x)) [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
p :: 0:s -> 0:s
fac :: 0:s -> 0:s

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

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

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

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

(c) The following functions are completely defined:

   times_2
   fac_1
   p_1
   plus_2

Due to the following rules being added:

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

And the following fresh constants: none

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

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

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

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(s(0)) -> 0 [1]
   fac(s(x)) -> times(fac(p(s(x))), s(x)) [1]
   fac(v0) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
fac :: 0:s -> 0:s

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

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

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

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

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(s(0), y) -> plus(0, y) [2]
   times(s(x), 0) -> plus(0, 0) [2]
   times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(s(0)) -> 0 [1]
   fac(s(s(x''))) -> times(fac(s(p(s(x'')))), s(s(x''))) [2]
   fac(s(0)) -> times(fac(0), s(0)) [2]
   fac(s(x)) -> times(fac(0), s(x)) [1]
   fac(v0) -> 0 [0]
   p(v0) -> 0 [0]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
p :: 0:s -> 0:s
fac :: 0:s -> 0:s

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

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

   0 => 0

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

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

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


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

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

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

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


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

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

   { plus }
   { p }
   { times }
   { fac }

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

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

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

Function symbols to be analyzed: {plus}, {p}, {times}, {fac}

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

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

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

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

Function symbols to be analyzed: {plus}, {p}, {times}, {fac}

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

(21) 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'

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

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

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

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

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

(23) 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'

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

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

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

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

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

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

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

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

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

   fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0
   fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0
   fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0
   fac(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(33) 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: 2*z*z' + z'

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

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

   fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0
   fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0
   fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0
   fac(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(35) 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: 8 + 4*z + 2*z*z' + z'

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

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

   fac(z) -{ 2 }-> times(fac(0), 1 + 0) :|: z = 1 + 0
   fac(z) -{ 1 }-> times(fac(0), 1 + (z - 1)) :|: z - 1 >= 0
   fac(z) -{ 2 + z }-> times(fac(1 + s2), 1 + (1 + (z - 2))) :|: s2 >= 0, s2 <= 1 + (z - 2), z - 2 >= 0
   fac(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> 0 :|: z = 1 + 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + (z - 2), z - 2 >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s :|: s >= 0, s <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> s' :|: s' >= 0, s' <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> s'' :|: s'' >= 0, s'' <= 0 + 0, z - 1 >= 0, z' = 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {fac}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  p: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z']

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

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

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

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

Function symbols to be analyzed: {fac}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  p: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {fac}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  p: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z']
  fac: runtime: ?, size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

Function symbols to be analyzed: {fac}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  p: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  times: runtime: O(n^2) [8 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + z']
  fac: runtime: INF, size: INF

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

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

   p(v0) -> null_p [0]
   fac(v0) -> null_fac [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]

And the following fresh constants:    null_p, null_fac, null_plus, null_times

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

(44)
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(x, s(y)) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(x, 0) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   p(s(0)) -> 0 [1]
   fac(s(x)) -> times(fac(p(s(x))), s(x)) [1]
   p(v0) -> null_p [0]
   fac(v0) -> null_fac [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]

The TRS has the following type information:
plus :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times
0 :: 0:s:null_p:null_fac:null_plus:null_times
s :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times
times :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times
p :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times
fac :: 0:s:null_p:null_fac:null_plus:null_times -> 0:s:null_p:null_fac:null_plus:null_times
null_p :: 0:s:null_p:null_fac:null_plus:null_times
null_fac :: 0:s:null_p:null_fac:null_plus:null_times
null_plus :: 0:s:null_p:null_fac:null_plus:null_times
null_times :: 0:s:null_p:null_fac:null_plus:null_times

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

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

   0 => 0
   null_p => 0
   null_fac => 0
   null_plus => 0
   null_times => 0

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

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

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

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, 0) -> c
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(z0, 0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   P(s(0)) -> c6
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
S tuples:
   PLUS(z0, 0) -> c
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(0, z0) -> c2
   TIMES(z0, 0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   P(s(0)) -> c6
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c, c1_1, c2, c3, c4_2, c5_1, c6, c7_3


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   TIMES(0, z0) -> c2
   P(s(0)) -> c6
   PLUS(z0, 0) -> c
   TIMES(z0, 0) -> c3

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
S tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
K tuples:none
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3


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

(51) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
We considered the (Usable) Rules:
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   plus(z0, 0) -> z0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
   times(0, z0) -> 0
   times(z0, 0) -> 0
And the Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FAC(x_1)) = 0
   POL(P(x_1)) = 0
   POL(PLUS(x_1, x_2)) = x_2
   POL(TIMES(x_1, x_2)) = x_1*x_2
   POL(c1(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(fac(x_1)) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_1 + [2]x_2
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [2]x_1*x_2

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
We considered the (Usable) Rules:
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   plus(z0, 0) -> z0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
   times(0, z0) -> 0
   times(z0, 0) -> 0
And the Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FAC(x_1)) = 0
   POL(P(x_1)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(fac(x_1)) = 0
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_1 + x_2
   POL(s(x_1)) = [2] + x_1
   POL(times(x_1, x_2)) = x_1*x_2

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0)))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(z0)) -> c7(TIMES(fac(p(s(z0))), s(z0)), FAC(p(s(z0))), P(s(z0))) by 
   FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0))))
   FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0))))
   FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0)))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0))))
   FAC(s(0)) -> c7(TIMES(fac(0), s(0)), FAC(p(s(0))), P(s(0)))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0))))
   FAC(s(0)) -> c7(FAC(p(s(0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0))))
   FAC(s(0)) -> c7(FAC(p(s(0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(s(z0))) -> c7(TIMES(fac(s(p(s(z0)))), s(s(z0))), FAC(p(s(s(z0)))), P(s(s(z0)))) by 
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(0)) -> c7(FAC(p(s(0))))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(0)) -> c7(FAC(p(s(0))))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_1, c7_3, c7_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(0)) -> c7(FAC(p(s(0)))) by 
   FAC(s(0)) -> c7(FAC(0))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(0)) -> c7(FAC(0))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(0)) -> c7(FAC(0))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_2, c7_1


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

(63) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FAC(s(0)) -> c7(FAC(0))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_2


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

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(x0))) -> c7(TIMES(times(fac(p(s(p(s(x0))))), s(p(s(x0)))), s(s(x0))), FAC(p(s(s(x0)))), P(s(s(x0)))) by FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_2


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(p(s(s(s(z0))))), P(s(s(s(z0))))) by FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0))))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_2


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(p(s(s(0)))), P(s(s(0)))) by FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0))))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_2, c7_3


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

(71) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(x0))) -> c7(FAC(p(s(s(x0)))), P(s(s(x0)))) by FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
----------------------------------------

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
K tuples:
   PLUS(z0, s(z1)) -> c1(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   PLUS_2, TIMES_2, P_1, FAC_1

Compound Symbols:   c1_1, c4_2, c5_1, c7_3, c7_2


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

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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
S tuples:
   P(s(s(z0))) -> c5(P(s(z0)))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, P_1, FAC_1, PLUS_2

Compound Symbols:   c4_2, c5_1, c7_3, c7_2, c1_1


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

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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))), P(s(s(0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, PLUS_2, P_1

Compound Symbols:   c4_2, c7_3, c7_2, c1_1, c5_1


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, PLUS_2, P_1

Compound Symbols:   c4_2, c7_3, c7_2, c1_1, c5_1


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(p(s(s(z0))))), P(s(s(s(z0))))) by FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1)))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, PLUS_2, P_1

Compound Symbols:   c4_2, c7_3, c7_2, c1_1, c5_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c1(PLUS(z0, s(y1))) by 
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0)))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, P_1, PLUS_2

Compound Symbols:   c4_2, c7_3, c7_2, c5_1, c1_1


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(p(s(0))))) by FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0)))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0)))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))), FAC(s(0)))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, P_1, PLUS_2

Compound Symbols:   c4_2, c7_3, c7_2, c5_1, c1_1


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, P_1, PLUS_2

Compound Symbols:   c4_2, c7_3, c7_2, c5_1, c1_1, c7_1


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FAC(x_1)) = [1] + x_1
   POL(P(x_1)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = x_2
   POL(c1(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(fac(x_1)) = x_1
   POL(p(x_1)) = 0
   POL(plus(x_1, x_2)) = x_1
   POL(s(x_1)) = 0
   POL(times(x_1, x_2)) = x_2

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   P(s(s(s(y0)))) -> c5(P(s(s(y0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, P_1, PLUS_2

Compound Symbols:   c4_2, c7_3, c7_2, c5_1, c1_1, c7_1


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

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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(z0, 0) -> 0
   times(s(z0), z1) -> plus(times(z0, z1), z1)
   p(s(s(z0))) -> s(p(s(z0)))
   p(s(0)) -> 0
   fac(s(z0)) -> times(fac(p(s(z0))), s(z0))
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
   P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0)))))
S tuples:
   FAC(s(s(z0))) -> c7(TIMES(times(fac(p(s(p(s(z0))))), s(p(s(z0)))), s(s(z0))), FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(z0))) -> c7(FAC(s(p(s(z0)))), P(s(s(z0))))
   FAC(s(s(s(z0)))) -> c7(TIMES(fac(s(s(p(s(z0))))), s(s(s(z0)))), FAC(s(s(p(s(z0))))), P(s(s(s(z0)))))
   P(s(s(s(s(y0))))) -> c5(P(s(s(s(y0)))))
K tuples:
   TIMES(s(z0), z1) -> c4(PLUS(times(z0, z1), z1), TIMES(z0, z1))
   PLUS(z0, s(s(s(y1)))) -> c1(PLUS(z0, s(s(y1))))
   FAC(s(s(0))) -> c7(TIMES(fac(s(0)), s(s(0))))
Defined Rule Symbols:   plus_2, times_2, p_1, fac_1

Defined Pair Symbols:   TIMES_2, FAC_1, PLUS_2, P_1

Compound Symbols:   c4_2, c7_3, c7_2, c1_1, c7_1, c5_1