KILLED
proof of input_JOMpMVynbq.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) InliningProof [UPPER BOUND(ID), 50 ms]
        (18) CpxRNTS
        (19) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 262 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 29 ms]
        (28) CpxRNTS
        (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 88 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
        (34) CpxRNTS
        (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 436 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 155 ms]
        (40) CpxRNTS
        (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 428 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
    (47) CompletionProof [UPPER BOUND(ID), 0 ms]
    (48) CpxTypedWeightedCompleteTrs
        (49) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
(51) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(52) CdtProblem
    (53) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 115 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem


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

(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:

   fac(0) -> 1
   fac(s(x)) -> *(s(x), fac(x))
   floop(0, y) -> y
   floop(s(x), y) -> floop(x, *(s(x), y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   1 -> s(0)
   fac(0) -> s(0)

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

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

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


The TRS R consists of the following rules:

   fac(0') -> 1'
   fac(s(x)) -> *'(s(x), fac(x))
   floop(0', y) -> y
   floop(s(x), y) -> floop(x, *'(s(x), y))
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(*'(x, y), x)
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   1' -> s(0')
   fac(0') -> s(0')

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

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

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


The TRS R consists of the following rules:

   fac(0) -> 1
   fac(s(x)) -> *(s(x), fac(x))
   floop(0, y) -> y
   floop(s(x), y) -> floop(x, *(s(x), y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   1 -> s(0)
   fac(0) -> s(0)

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

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

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


The TRS R consists of the following rules:

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

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

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

1 => 1'
* => 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:

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

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

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

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

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

The TRS has the following type information:
fac :: 0:s -> 0:s
0 :: 0:s
1' :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
floop :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

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

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

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

   floop_2

(c) The following functions are completely defined:

   times_2
   fac_1
   plus_2
   1'

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

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

The TRS has the following type information:
fac :: 0:s -> 0:s
0 :: 0:s
1' :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
floop :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

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

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

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

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

The TRS has the following type information:
fac :: 0:s -> 0:s
0 :: 0:s
1' :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
floop :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

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

   0 => 0

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

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

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


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

(17) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

   1' -{ 1 }-> 1 + 0 :|: 

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

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

   1' -{ 1 }-> 1 + 0 :|: 
   fac(z) -{ 2 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0
   fac(z) -{ 3 }-> times(1 + 0, 1 + 0) :|: z = 1 + 0
   fac(z) -{ 2 }-> times(1 + (1 + x'), times(1 + x', fac(x'))) :|: x' >= 0, z = 1 + (1 + x')
   fac(z) -{ 1 }-> 1 + 0 :|: z = 0
   fac(z) -{ 2 }-> 1 + 0 :|: z = 0
   floop(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   floop(z, z') -{ 2 }-> floop(x, plus(times(1 + x, y'), 1 + x)) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   floop(z, z') -{ 2 }-> floop(x, 0) :|: x >= 0, z = 1 + x, z' = 0
   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''), x), x) :|: z' = 1 + (1 + y''), x >= 0, y'' >= 0, z = x
   times(z, z') -{ 2 }-> plus(0, x) :|: x >= 0, z' = 1 + 0, z = x
   times(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0


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

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

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

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


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

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

   { plus }
   { 1' }
   { times }
   { fac }
   { floop }

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

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

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

Function symbols to be analyzed: {plus}, {1'}, {times}, {fac}, {floop}

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

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

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

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

Function symbols to be analyzed: {plus}, {1'}, {times}, {fac}, {floop}

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

(25) 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'

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

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

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

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

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

(27) 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'

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

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

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

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

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

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: 1'
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:

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

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

   1' -{ 1 }-> 1 + 0 :|: 
   fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0
   fac(z) -{ 1 }-> 1 + 0 :|: z = 0
   fac(z) -{ 2 }-> 1 + 0 :|: z = 0
   floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0
   floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(43) 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: ?

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

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

   1' -{ 1 }-> 1 + 0 :|: 
   fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0
   fac(z) -{ 1 }-> 1 + 0 :|: z = 0
   fac(z) -{ 2 }-> 1 + 0 :|: z = 0
   floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0
   floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

(45) 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: ?

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

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

   1' -{ 1 }-> 1 + 0 :|: 
   fac(z) -{ 13 }-> s'' :|: s'' >= 0, s'' <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 14 }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 2 * ((1 + 0) * (1 + 0)), z = 1 + 0
   fac(z) -{ 2 }-> times(1 + (1 + (z - 2)), times(1 + (z - 2), fac(z - 2))) :|: z - 2 >= 0
   fac(z) -{ 1 }-> 1 + 0 :|: z = 0
   fac(z) -{ 2 }-> 1 + 0 :|: z = 0
   floop(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   floop(z, z') -{ 3 + 2*z*z' + 4*z' }-> floop(z - 1, s2) :|: s1 >= 0, s1 <= 1 + (z - 1) + 2 * ((z' - 1) * (1 + (z - 1))), s2 >= 0, s2 <= s1 + (1 + (z - 1)), z - 1 >= 0, z' - 1 >= 0
   floop(z, z') -{ 2 }-> floop(z - 1, 0) :|: z - 1 >= 0, z' = 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 >= 0, z' = 1 + 0
   times(z, z') -{ -1*z + 2*z*z' + 4*z' }-> s5 :|: s3 >= 0, s3 <= z + 2 * ((z' - 2) * z), s4 >= 0, s4 <= s3 + z, s5 >= 0, s5 <= s4 + z, z >= 0, z' - 2 >= 0
   times(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0

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

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

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

And the following fresh constants: none

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

(48)
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:

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

The TRS has the following type information:
fac :: 0:s -> 0:s
0 :: 0:s
1' :: 0:s
s :: 0:s -> 0:s
times :: 0:s -> 0:s -> 0:s
floop :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s

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

(49) 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

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

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

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

Only complete derivations are relevant for the runtime complexity.

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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   floop(0, z0) -> z0
   floop(s(z0), z1) -> floop(z0, *(s(z0), z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   1 -> s(0)
Tuples:
   FAC(0) -> c(1')
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FAC(0) -> c2
   FLOOP(0, z0) -> c3
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, 0) -> c5
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   1' -> c9
S tuples:
   FAC(0) -> c(1')
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FAC(0) -> c2
   FLOOP(0, z0) -> c3
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, 0) -> c5
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, 0) -> c7
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   1' -> c9
K tuples:none
Defined Rule Symbols:   fac_1, floop_2, *_2, +_2, 1

Defined Pair Symbols:   FAC_1, FLOOP_2, *'_2, +'_2, 1'

Compound Symbols:   c_1, c1_2, c2, c3, c4_2, c5, c6_2, c7, c8_1, c9


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

(53) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   FAC(0) -> c2
   +'(z0, 0) -> c7
   1' -> c9
   *'(z0, 0) -> c5
   FAC(0) -> c(1')
   FLOOP(0, z0) -> c3

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   floop(0, z0) -> z0
   floop(s(z0), z1) -> floop(z0, *(s(z0), z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   1 -> s(0)
Tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
S tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   fac_1, floop_2, *_2, +_2, 1

Defined Pair Symbols:   FAC_1, FLOOP_2, *'_2, +'_2

Compound Symbols:   c1_2, c4_2, c6_2, c8_1


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

(55) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   floop(0, z0) -> z0
   floop(s(z0), z1) -> floop(z0, *(s(z0), z1))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
S tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FAC_1, FLOOP_2, *'_2, +'_2

Compound Symbols:   c1_2, c4_2, c6_2, c8_1


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
We considered the (Usable) Rules:none
And the Tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = [1] + x_1
   POL(*'(x_1, x_2)) = 0
   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(1) = [1]
   POL(FAC(x_1)) = x_1
   POL(FLOOP(x_1, x_2)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(fac(x_1)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FAC_1, FLOOP_2, *'_2, +'_2

Compound Symbols:   c1_2, c4_2, c6_2, c8_1


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0)) by 
   FAC(s(0)) -> c1(*'(s(0), 1), FAC(0))
   FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0)))
   FAC(s(0)) -> c1(*'(s(0), s(0)), FAC(0))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), 1), FAC(0))
   FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0)))
   FAC(s(0)) -> c1(*'(s(0), s(0)), FAC(0))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_2


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0)))
   FAC(s(0)) -> c1(*'(s(0), 1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_2, c1_1


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(s(z0))) -> c1(*'(s(s(z0)), *(s(z0), fac(z0))), FAC(s(z0))) by 
   FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), 1)), FAC(s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), s(0))), FAC(s(0)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), 1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), 1)), FAC(s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c1(*'(s(s(0)), *(s(0), s(0))), FAC(s(0)))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_1, c1_2


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

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), 1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1)))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_1, c1_2, c_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FAC(s(0)) -> c1(*'(s(0), 1)) by 
   FAC(s(0)) -> c1(*'(s(0), s(0)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1)))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_1, c1_2, c_1


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), 1))) by FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(z1)) -> c8(+'(z0, z1))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, +'_2, FAC_1

Compound Symbols:   c4_2, c6_2, c8_1, c1_1, c1_2, c_1


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(z1)) -> c8(+'(z0, z1)) by 
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, FAC_1, +'_2

Compound Symbols:   c4_2, c6_2, c1_1, c1_2, c_1, c8_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), *(s(0), s(0))))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
   FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, FAC_1, +'_2

Compound Symbols:   c4_2, c6_2, c1_1, c1_2, c_1, c8_1


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
   FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, FAC_1, +'_2

Compound Symbols:   c4_2, c6_2, c1_1, c1_2, c_1, c8_1


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(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0)))) by FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0))))
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(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
   FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0))))
   FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c8(+'(z0, s(y1)))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, FAC_1, +'_2

Compound Symbols:   c4_2, c6_2, c1_1, c1_2, c_1, c8_1


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(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(s(y1))) -> c8(+'(z0, s(y1))) by 
   +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1))))

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(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fac(0) -> 1
   fac(s(z0)) -> *(s(z0), fac(z0))
   fac(0) -> s(0)
   1 -> s(0)
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   FAC(s(0)) -> c1(*'(s(0), s(0)))
   FAC(s(s(s(z0)))) -> c1(*'(s(s(s(z0))), *(s(s(z0)), *(s(z0), fac(z0)))), FAC(s(s(z0))))
   FAC(s(s(0))) -> c(FAC(s(0)))
   FAC(s(s(0))) -> c(*'(s(s(0)), +(*(s(0), 0), s(0))))
   FAC(s(s(0))) -> c(*'(s(s(0)), s(+(*(s(0), 0), 0))))
   +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c6(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c8(+'(z0, s(s(y1))))
K tuples:
   FAC(s(z0)) -> c1(*'(s(z0), fac(z0)), FAC(z0))
   FLOOP(s(z0), z1) -> c4(FLOOP(z0, *(s(z0), z1)), *'(s(z0), z1))
Defined Rule Symbols:   fac_1, 1, *_2, +_2

Defined Pair Symbols:   FLOOP_2, *'_2, FAC_1, +'_2

Compound Symbols:   c4_2, c6_2, c1_1, c1_2, c_1, c8_1