KILLED
proof of input_AvQP6Lb4VN.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 333 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 76 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 248 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 148 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 231 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 507 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
        (44) CpxRNTS
        (45) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 950 ms]
        (48) CpxRNTS
        (49) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
    (51) CompletionProof [UPPER BOUND(ID), 0 ms]
    (52) CpxTypedWeightedCompleteTrs
        (53) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (54) CpxRNTS
(55) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(56) CdtProblem
    (57) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 32 ms]
    (70) CdtProblem
    (71) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 30 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 33 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 31 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
    (104) CdtProblem
    (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
    (106) CdtProblem
    (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms]
    (108) CdtProblem
    (109) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 65 ms]
    (110) CdtProblem
    (111) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   fact(x) -> iffact(x, ge(x, s(s(0))))
   iffact(x, true) -> *(x, fact(-(x, s(0))))
   iffact(x, false) -> 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:

   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(x, y))
   *'(x, 0') -> 0'
   *'(x, s(y)) -> +'(*'(x, y), x)
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   -(x, 0') -> x
   -(s(x), s(y)) -> -(x, y)
   fact(x) -> iffact(x, ge(x, s(s(0'))))
   iffact(x, true) -> *'(x, fact(-(x, s(0'))))
   iffact(x, false) -> 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:

   +(x, 0) -> x
   +(x, s(y)) -> s(+(x, y))
   *(x, 0) -> 0
   *(x, s(y)) -> +(*(x, y), x)
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   fact(x) -> iffact(x, ge(x, s(s(0))))
   iffact(x, true) -> *(x, fact(-(x, s(0))))
   iffact(x, false) -> 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:

   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]
   *(x, 0) -> 0 [1]
   *(x, s(y)) -> +(*(x, y), x) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   -(x, 0) -> x [1]
   -(s(x), s(y)) -> -(x, y) [1]
   fact(x) -> iffact(x, ge(x, s(s(0)))) [1]
   iffact(x, true) -> *(x, fact(-(x, s(0)))) [1]
   iffact(x, false) -> s(0) [1]

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

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

+ => plus
* => times
- => minus

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

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


The TRS R consists of the following rules:

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   fact(x) -> iffact(x, ge(x, s(s(0)))) [1]
   iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1]
   iffact(x, false) -> 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:

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   fact(x) -> iffact(x, ge(x, s(s(0)))) [1]
   iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1]
   iffact(x, false) -> s(0) [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
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
minus :: 0:s -> 0:s -> 0:s
fact :: 0:s -> 0:s
iffact :: 0:s -> true:false -> 0:s

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

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

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

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

(c) The following functions are completely defined:

   times_2
   ge_2
   fact_1
   minus_2
   plus_2
   iffact_2

Due to the following rules being added:

   minus(v0, v1) -> 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:

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   times(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   fact(x) -> iffact(x, ge(x, s(s(0)))) [1]
   iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1]
   iffact(x, false) -> s(0) [1]
   minus(v0, v1) -> 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
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
minus :: 0:s -> 0:s -> 0:s
fact :: 0:s -> 0:s
iffact :: 0:s -> true:false -> 0:s

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

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

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

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

   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   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]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   fact(0) -> iffact(0, false) [2]
   fact(s(x')) -> iffact(s(x'), ge(x', s(0))) [2]
   iffact(s(x''), true) -> times(s(x''), fact(minus(x'', 0))) [2]
   iffact(x, true) -> times(x, fact(0)) [1]
   iffact(x, false) -> s(0) [1]
   minus(v0, v1) -> 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
ge :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
minus :: 0:s -> 0:s -> 0:s
fact :: 0:s -> 0:s
iffact :: 0:s -> true:false -> 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
   true => 1
   false => 0

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

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

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


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

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

   { minus }
   { ge }
   { plus }
   { times }
   { fact, iffact }

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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: {minus}, {ge}, {plus}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: ?, size: O(n^1) [z]

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 2 }-> times(1 + (z - 1), fact(minus(z - 1, 0))) :|: z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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: {ge}, {plus}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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: {ge}, {plus}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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: {ge}, {plus}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: ?, size: O(1) [1]

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 2 }-> iffact(1 + (z - 1), ge(z - 1, 1 + 0)) :|: z - 1 >= 0
   ge(z, z') -{ 1 }-> ge(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(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) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(35) 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'

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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}, {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: ?, size: O(n^1) [z + z']

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

(37) 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'

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, 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: {times}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [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) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 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}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']

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

(41) 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'

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 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}, {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  times: runtime: ?, size: O(n^2) [z + 2*z*z']

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

(43) 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'

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

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

   fact(z) -{ 2 }-> iffact(0, 0) :|: z = 0
   fact(z) -{ 5 }-> iffact(1 + (z - 1), s1) :|: s1 >= 0, s1 <= 1, z - 1 >= 0
   ge(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z - 1 >= 0, z' - 1 >= 0
   ge(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   ge(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   iffact(z, z') -{ 1 }-> times(z, fact(0)) :|: z >= 0, z' = 1
   iffact(z, z') -{ 3 }-> times(1 + (z - 1), fact(s')) :|: s' >= 0, s' <= z - 1, z' = 1, z - 1 >= 0
   iffact(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   times(z, z') -{ 3 + z }-> s3 :|: s3 >= 0, s3 <= 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: {fact,iffact}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  ge: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  times: runtime: O(n^2) [4 + z + 2*z*z' + 4*z'], size: O(n^2) [z + 2*z*z']

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

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

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

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

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

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

(47) 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: ?

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

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

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

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

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

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

(49) 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: ?

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

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

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

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

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

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

   minus(v0, v1) -> null_minus [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   ge(v0, v1) -> null_ge [0]
   iffact(v0, v1) -> null_iffact [0]

And the following fresh constants:    null_minus, null_plus, null_times, null_ge, null_iffact

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

(52)
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(x, 0) -> 0 [1]
   times(x, s(y)) -> plus(times(x, y), x) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   fact(x) -> iffact(x, ge(x, s(s(0)))) [1]
   iffact(x, true) -> times(x, fact(minus(x, s(0)))) [1]
   iffact(x, false) -> s(0) [1]
   minus(v0, v1) -> null_minus [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   ge(v0, v1) -> null_ge [0]
   iffact(v0, v1) -> null_iffact [0]

The TRS has the following type information:
plus :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact
0 :: 0:s:null_minus:null_plus:null_times:null_iffact
s :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact
times :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact
ge :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> true:false:null_ge
true :: true:false:null_ge
false :: true:false:null_ge
minus :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact
fact :: 0:s:null_minus:null_plus:null_times:null_iffact -> 0:s:null_minus:null_plus:null_times:null_iffact
iffact :: 0:s:null_minus:null_plus:null_times:null_iffact -> true:false:null_ge -> 0:s:null_minus:null_plus:null_times:null_iffact
null_minus :: 0:s:null_minus:null_plus:null_times:null_iffact
null_plus :: 0:s:null_minus:null_plus:null_times:null_iffact
null_times :: 0:s:null_minus:null_plus:null_times:null_iffact
null_ge :: true:false:null_ge
null_iffact :: 0:s:null_minus:null_plus:null_times:null_iffact

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

(53) 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
   true => 2
   false => 1
   null_minus => 0
   null_plus => 0
   null_times => 0
   null_ge => 0
   null_iffact => 0

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

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

Only complete derivations are relevant for the runtime complexity.

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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, 0) -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(z0, 0) -> c7
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0))))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   IFFACT(z0, false) -> c11
S tuples:
   +'(z0, 0) -> c
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, 0) -> c2
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(z0, 0) -> c7
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0))))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   IFFACT(z0, false) -> c11
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c, c1_1, c2, c3_2, c4, c5, c6_1, c7, c8_1, c9_2, c10_3, c11


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

(57) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   IFFACT(z0, false) -> c11
   *'(z0, 0) -> c2
   -'(z0, 0) -> c7
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   +'(z0, 0) -> c

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0))))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0))))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_2, c10_3


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FACT(z0) -> c9(IFFACT(z0, ge(z0, s(s(0)))), GE(z0, s(s(0)))) by 
   FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0))))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0))))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0))))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FACT(0) -> c9(IFFACT(0, false), GE(0, s(s(0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0)))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2


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

(63) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFFACT(z0, true) -> c10(*'(z0, fact(-(z0, s(0)))), FACT(-(z0, s(0))), -'(z0, s(0))) by 
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0))))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_2, c10_3


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FACT(s(z0)) -> c9(IFFACT(s(z0), ge(z0, s(0))), GE(s(z0), s(s(0)))) by 
   FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(0)) -> c9(IFFACT(s(0), false), GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
K tuples:none
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1


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

(69) 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(0)) -> c9(GE(s(0), s(s(0))))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
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)) = x_1
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(FACT(x_1)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = x_1
   POL(true) = 0

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1


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

(71) 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(x0)) -> c9(GE(s(x0), s(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
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)) = x_1
   POL(+(x_1, x_2)) = x_1
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = 0
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = [1]
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = [1] + x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(true) = 0

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFFACT(x0, true) -> c10(*'(x0, iffact(-(x0, s(0)), ge(-(x0, s(0)), s(s(0))))), FACT(-(x0, s(0))), -'(x0, s(0))) by 
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, IFFACT_2, FACT_1

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c10_3, c9_2, c9_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFFACT(s(z0), true) -> c10(*'(s(z0), fact(-(z0, 0))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by 
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_2, c9_1, c10_3


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), ge(z0, 0)), GE(s(s(z0)), s(s(0)))) by FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(s(z0), s(0)), ge(-(z0, 0), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by 
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(83) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(x0), true) -> c10(*'(s(x0), iffact(-(x0, 0), ge(-(x0, 0), s(s(0))))), FACT(-(s(x0), s(0))), -'(s(x0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = [1]
   POL(FACT(x_1)) = [1] + x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = [1] + x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c10_3, c9_2


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), fact(z0)), FACT(-(s(z0), s(0))), -'(s(z0), s(0))) by 
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(s(x0)), s(0)), ge(-(s(x0), 0), s(s(0))))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3


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

(93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
We considered the (Usable) Rules:
   -(s(z0), s(z1)) -> -(z0, z1)
   -(z0, 0) -> z0
And the Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   +'(z0, s(z1)) -> c1(+'(z0, z1))
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   +'_2, *'_2, GE_2, -'_2, FACT_1, IFFACT_2

Compound Symbols:   c1_1, c3_2, c6_1, c8_1, c9_1, c9_2, c10_3


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

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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(s(z0), s(0)), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) by 
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0))) by IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), fact(s(x0))), FACT(-(s(s(x0)), s(0))), -'(s(s(x0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(103) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
We considered the (Usable) Rules:
   -(z0, 0) -> z0
And the Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = x_2
   POL(0) = 0
   POL(FACT(x_1)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
We considered the (Usable) Rules:
   -(z0, 0) -> z0
And the Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(107) 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(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
We considered the (Usable) Rules:
   -(z0, 0) -> z0
And the Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
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(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = [1] + x_1
   POL(GE(x_1, x_2)) = 0
   POL(IFFACT(x_1, x_2)) = x_1 + x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(109) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
We considered the (Usable) Rules:
   -(z0, 0) -> z0
And the Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(*(x_1, x_2)) = 0
   POL(*'(x_1, x_2)) = 0
   POL(+(x_1, x_2)) = [3] + [3]x_1 + [3]x_2
   POL(+'(x_1, x_2)) = 0
   POL(-(x_1, x_2)) = x_1
   POL(-'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FACT(x_1)) = [2] + [2]x_1
   POL(GE(x_1, x_2)) = x_2
   POL(IFFACT(x_1, x_2)) = [2]x_1 + [3]x_2
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2, x_3)) = x_1 + x_2 + x_3
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(fact(x_1)) = [3]
   POL(false) = [3]
   POL(ge(x_1, x_2)) = 0
   POL(iffact(x_1, x_2)) = [3]
   POL(s(x_1)) = [1] + x_1
   POL(true) = 0

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(111) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(z0), -'(s(z0), s(0))) by 
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, GE_2, -'_2, FACT_1, IFFACT_2, +'_2

Compound Symbols:   c3_2, c6_1, c8_1, c9_1, c9_2, c10_3, c1_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c6(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(z0), true) -> c10(*'(s(z0), iffact(-(z0, 0), ge(-(z0, 0), s(s(0))))), FACT(-(z0, 0)), -'(s(z0), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, -'_2, FACT_1, IFFACT_2, +'_2, GE_2

Compound Symbols:   c3_2, c8_1, c9_1, c9_2, c10_3, c1_1, c6_1


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

(115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FACT(s(0)) -> c9(GE(s(0), s(s(0))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, -'_2, FACT_1, IFFACT_2, +'_2, GE_2

Compound Symbols:   c3_2, c8_1, c9_1, c9_2, c10_3, c1_1, c6_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0)))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   -'(s(z0), s(z1)) -> c8(-'(z0, z1))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, -'_2, FACT_1, +'_2, IFFACT_2, GE_2

Compound Symbols:   c3_2, c8_1, c9_1, c9_2, c1_1, c10_3, c6_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(z0), s(z1)) -> c8(-'(z0, z1)) by 
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)), -'(s(s(x0)), s(0)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)), -'(s(s(z0)), s(0)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)), -'(s(s(x0)), s(0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2, IFFACT_2, GE_2, -'_2

Compound Symbols:   c3_2, c9_1, c9_2, c1_1, c10_3, c6_1, c8_1


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

(121) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing tuple parts
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2

Compound Symbols:   c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2

Compound Symbols:   c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0))) by IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   FACT(s(x0)) -> c9(GE(s(x0), s(s(0))))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2

Compound Symbols:   c3_2, c9_1, c9_2, c1_1, c6_1, c8_1, c10_2


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

(127) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FACT(s(x0)) -> c9(GE(s(x0), s(s(0)))) by 
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(y1))) -> c1(+'(z0, s(y1)))
K tuples:
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, +'_2, GE_2, -'_2, IFFACT_2

Compound Symbols:   c3_2, c9_2, c1_1, c6_1, c8_1, c10_2, c9_1


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

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

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
K tuples:
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, GE_2, -'_2, IFFACT_2, +'_2

Compound Symbols:   c3_2, c9_2, c6_1, c8_1, c10_2, c9_1, c1_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) by 
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
K tuples:
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(-(s(z0), 0)), -'(s(s(z0)), s(0)))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(s(x0)), s(0)), s(s(0))))), FACT(-(s(x0), 0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(-(s(z0), 0)))
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, -'_2, IFFACT_2, +'_2, GE_2

Compound Symbols:   c3_2, c9_2, c8_1, c10_2, c9_1, c1_1, c6_1


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

(133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FACT(s(s(y0))) -> c9(GE(s(s(y0)), s(s(0))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
K tuples:
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true), GE(s(s(z0)), s(s(0))))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, FACT_1, -'_2, IFFACT_2, +'_2, GE_2

Compound Symbols:   c3_2, c9_2, c8_1, c10_2, c1_1, c6_1


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

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
K tuples:
   -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, -'_2, IFFACT_2, +'_2, GE_2, FACT_1

Compound Symbols:   c3_2, c8_1, c10_2, c1_1, c6_1, c9_1


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

(137) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace -'(s(s(y0)), s(s(y1))) -> c8(-'(s(y0), s(y1))) by 
   -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
   *(z0, 0) -> 0
   *(z0, s(z1)) -> +(*(z0, z1), z0)
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   -(z0, 0) -> z0
   -(s(z0), s(z1)) -> -(z0, z1)
   fact(z0) -> iffact(z0, ge(z0, s(s(0))))
   iffact(z0, true) -> *(z0, fact(-(z0, s(0))))
   iffact(z0, false) -> s(0)
Tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   IFFACT(s(s(x0)), true) -> c10(*'(s(s(x0)), iffact(-(s(x0), 0), ge(-(s(x0), 0), s(s(0))))), FACT(s(x0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(s(z0)), s(0)), ge(-(s(z0), 0), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), iffact(-(s(z0), 0), ge(-(s(s(z0)), s(0)), s(s(0))))), FACT(s(z0)))
   IFFACT(s(s(z0)), true) -> c10(*'(s(s(z0)), fact(s(z0))), FACT(s(z0)))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1))))
S tuples:
   *'(z0, s(z1)) -> c3(+'(*(z0, z1), z0), *'(z0, z1))
   +'(z0, s(s(s(y1)))) -> c1(+'(z0, s(s(y1))))
K tuples:
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   FACT(s(s(z0))) -> c9(IFFACT(s(s(z0)), true))
   -'(s(s(s(y0))), s(s(s(y1)))) -> c8(-'(s(s(y0)), s(s(y1))))
Defined Rule Symbols:   +_2, *_2, ge_2, -_2, fact_1, iffact_2

Defined Pair Symbols:   *'_2, IFFACT_2, +'_2, GE_2, FACT_1, -'_2

Compound Symbols:   c3_2, c10_2, c1_1, c6_1, c9_1, c8_1