KILLED
proof of input_9mBoYnXA0p.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 251 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 68 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 422 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 136 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 913 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 4 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) CdtRhsSimplificationProcessorProof [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) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 131 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 1 ms]
    (64) CdtProblem
    (65) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) 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:

   p(s(x)) -> x
   fact(0) -> s(0)
   fact(s(x)) -> *(s(x), fact(p(s(x))))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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:

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

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:

   p(s(x)) -> x
   fact(0) -> s(0)
   fact(s(x)) -> *(s(x), fact(p(s(x))))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))

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:

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

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

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

* => times
+ => plus

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

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


The TRS R consists of the following rules:

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

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

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

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

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

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

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

(c) The following functions are completely defined:

   times_2
   fact_1
   p_1
   plus_2

Due to the following rules being added:

   p(v0) -> 0 [0]

And the following fresh constants: none

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

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

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

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

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

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:

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

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

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

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

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

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


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

(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:

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


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

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

   { plus }
   { p }
   { times }
   { fact }

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

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

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

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

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

(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:

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(25) 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'

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

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

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

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

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

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

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

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

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

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

(29) 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

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

   fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0
   fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0
   fact(z) -{ 1 }-> 1 + 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0
   fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0
   fact(z) -{ 1 }-> 1 + 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 1 }-> times(1 + (z - 1), fact(0)) :|: z - 1 >= 0
   fact(z) -{ 2 }-> times(1 + (z - 1), fact(z - 1)) :|: z - 1 >= 0
   fact(z) -{ 1 }-> 1 + 0 :|: z = 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 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') -{ 4*z + 2*z*z' + -1*z' }-> s2 :|: s'' >= 0, s'' <= z' + 2 * (z' * (z - 2)), s1 >= 0, s1 <= s'' + z', s2 >= 0, s2 <= s1 + z', z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

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

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

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

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

(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:

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

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

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_p => 0
   null_fact => 0
   null_times => 0
   null_plus => 0

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

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

   fact(z) -{ 1 }-> times(1 + x, fact(p(1 + x))) :|: x >= 0, z = 1 + x
   fact(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fact(z) -{ 1 }-> 1 + 0 :|: z = 0
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   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') -{ 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:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   P(s(z0)) -> c
   FACT(0) -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0, z0) -> c3
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   P(s(z0)) -> c
   FACT(0) -> c1
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(0, z0) -> c3
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, 0) -> c5
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   P_1, FACT_1, *'_2, +'_2

Compound Symbols:   c, c1, c2_3, c3, c4_2, c5, c6_1


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   FACT(0) -> c1
   *'(0, z0) -> c3
   P(s(z0)) -> c
   +'(z0, 0) -> c5

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
S tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))), P(s(z0)))
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   FACT_1, *'_2, +'_2

Compound Symbols:   c2_3, c4_2, c6_1


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))))
S tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, +'_2, FACT_1

Compound Symbols:   c4_2, c6_1, c2_2


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FACT(s(z0)) -> c2(*'(s(z0), fact(p(s(z0)))), FACT(p(s(z0)))) by 
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0))))
S tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0))))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, +'_2, FACT_1

Compound Symbols:   c4_2, c6_1, c2_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(p(s(z0)))) by 
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
S tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
K tuples:none
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, +'_2, FACT_1

Compound Symbols:   c4_2, c6_1, c2_2


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

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
We considered the (Usable) Rules:none
And the Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = x_2
   POL(*'(x_1, x_2)) = [2]
   POL(+(x_1, x_2)) = [3] + [3]x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [2]
   POL(FACT(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(fact(x_1)) = [1] + [3]x_1
   POL(p(x_1)) = [1]
   POL(s(x_1)) = [3] + x_1

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
S tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, +'_2, FACT_1

Compound Symbols:   c4_2, c6_1, c2_2


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [2]x_1
   POL(*'(x_1, x_2)) = [2] + [2]x_1
   POL(+(x_1, x_2)) = [1] + x_2 + x_2^2
   POL(+'(x_1, x_2)) = 0
   POL(0) = [2]
   POL(FACT(x_1)) = [2]x_1^2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(fact(x_1)) = [2] + [2]x_1^2
   POL(p(x_1)) = 0
   POL(s(x_1)) = [2] + x_1

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   +'(z0, s(z1)) -> c6(+'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
S tuples:
   +'(z0, s(z1)) -> c6(+'(z0, z1))
K tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, +'_2, FACT_1

Compound Symbols:   c4_2, c6_1, c2_2


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(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(z1)) -> c6(+'(z0, z1)) by 
   +'(z0, s(s(y1))) -> c6(+'(z0, s(y1)))

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(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
   +'(z0, s(s(y1))) -> c6(+'(z0, s(y1)))
S tuples:
   +'(z0, s(s(y1))) -> c6(+'(z0, s(y1)))
K tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2

Compound Symbols:   c4_2, c2_2, c6_1


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(65) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(s(y1))) -> c6(+'(z0, s(y1))) by 
   +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(s(z0)) -> z0
   fact(0) -> s(0)
   fact(s(z0)) -> *(s(z0), fact(p(s(z0))))
   *(0, z0) -> 0
   *(s(z0), z1) -> +(*(z0, z1), z1)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
   +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(s(y1)))) -> c6(+'(z0, s(s(y1))))
K tuples:
   FACT(s(z0)) -> c2(*'(s(z0), fact(z0)), FACT(z0))
   *'(s(z0), z1) -> c4(+'(*(z0, z1), z1), *'(z0, z1))
Defined Rule Symbols:   p_1, fact_1, *_2, +_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2

Compound Symbols:   c4_2, c2_2, c6_1