KILLED
proof of input_Ik5XX1y15n.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 35 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 427 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 303 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 506 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 198 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 5099 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 458 ms]
        (42) CpxRNTS
    (43) CompletionProof [UPPER BOUND(ID), 33 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 4 ms]
        (46) CpxRNTS
(47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(48) CdtProblem
    (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 20 ms]
    (66) CdtProblem
    (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 23 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (76) CdtProblem
    (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (80) CdtProblem
    (81) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRewritingProof [BOTH BOUNDS(ID, ID), 5 ms]
    (116) CdtProblem
    (117) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms]
    (120) CdtProblem
    (121) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) 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:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   log(x, 0) -> baseError
   log(x, s(0)) -> baseError
   log(0, s(s(b))) -> logZeroError
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0)
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
   if(true, x, b, y, z) -> z
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

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:

   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   log(x, 0') -> baseError
   log(x, s(0')) -> baseError
   log(0', s(s(b))) -> logZeroError
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0'), 0')
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
   if(true, x, b, y, z) -> z
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

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:

   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   log(x, 0) -> baseError
   log(x, s(0)) -> baseError
   log(0, s(s(b))) -> logZeroError
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0)
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z)
   if(true, x, b, y, z) -> z
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z))

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:

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   log(x, 0) -> baseError [1]
   log(x, s(0)) -> baseError [1]
   log(0, s(s(b))) -> logZeroError [1]
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1]
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1]
   if(true, x, b, y, z) -> z [1]
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1]

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

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

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   log(x, 0) -> baseError [1]
   log(x, s(0)) -> baseError [1]
   log(0, s(s(b))) -> logZeroError [1]
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1]
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1]
   if(true, x, b, y, z) -> z [1]
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1]

The TRS has the following type information:
le :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> false:true
s :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError
0 :: s:0:baseError:logZeroError
false :: false:true
true :: false:true
plus :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError
times :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError
log :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError
baseError :: s:0:baseError:logZeroError
logZeroError :: s:0:baseError:logZeroError
loop :: s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError
if :: false:true -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError -> s:0:baseError:logZeroError

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

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

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

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

   log_2
   loop_4
   if_5

(c) The following functions are completely defined:

   times_2
   le_2
   plus_2

Due to the following rules being added:

   times(v0, v1) -> null_times [0]
   le(v0, v1) -> null_le [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_times, null_le, null_plus

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

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

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   log(x, 0) -> baseError [1]
   log(x, s(0)) -> baseError [1]
   log(0, s(s(b))) -> logZeroError [1]
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1]
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1]
   if(true, x, b, y, z) -> z [1]
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1]
   times(v0, v1) -> null_times [0]
   le(v0, v1) -> null_le [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
le :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> false:true:null_le
s :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
0 :: s:0:baseError:logZeroError:null_times:null_plus
false :: false:true:null_le
true :: false:true:null_le
plus :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
times :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
log :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
baseError :: s:0:baseError:logZeroError:null_times:null_plus
logZeroError :: s:0:baseError:logZeroError:null_times:null_plus
loop :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
if :: false:true:null_le -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
null_times :: s:0:baseError:logZeroError:null_times:null_plus
null_le :: false:true:null_le
null_plus :: s:0:baseError:logZeroError:null_times:null_plus

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

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

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

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> plus(y, 0) [2]
   times(s(s(x')), y) -> plus(y, plus(y, times(x', y))) [2]
   times(s(x), y) -> plus(y, null_times) [1]
   log(x, 0) -> baseError [1]
   log(x, s(0)) -> baseError [1]
   log(0, s(s(b))) -> logZeroError [1]
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1]
   loop(0, s(s(b)), s(y), z) -> if(true, 0, s(s(b)), s(y), z) [2]
   loop(s(x''), s(s(b)), s(y), z) -> if(le(x'', y), s(x''), s(s(b)), s(y), z) [2]
   loop(x, s(s(b)), s(y), z) -> if(null_le, x, s(s(b)), s(y), z) [1]
   if(true, x, b, y, z) -> z [1]
   if(false, x, 0, y, z) -> loop(x, 0, 0, s(z)) [2]
   if(false, x, s(x1), y, z) -> loop(x, s(x1), plus(y, times(x1, y)), s(z)) [2]
   if(false, x, b, y, z) -> loop(x, b, null_times, s(z)) [1]
   times(v0, v1) -> null_times [0]
   le(v0, v1) -> null_le [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
le :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> false:true:null_le
s :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
0 :: s:0:baseError:logZeroError:null_times:null_plus
false :: false:true:null_le
true :: false:true:null_le
plus :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
times :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
log :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
baseError :: s:0:baseError:logZeroError:null_times:null_plus
logZeroError :: s:0:baseError:logZeroError:null_times:null_plus
loop :: s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
if :: false:true:null_le -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus -> s:0:baseError:logZeroError:null_times:null_plus
null_times :: s:0:baseError:logZeroError:null_times:null_plus
null_le :: false:true:null_le
null_plus :: s:0:baseError:logZeroError:null_times:null_plus

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

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

   0 => 0
   false => 1
   true => 2
   baseError => 1
   logZeroError => 2
   null_times => 0
   null_le => 0
   null_plus => 0

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = b, b >= 0, z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(x, b, 0, 1 + z) :|: z1 = b, b >= 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(x, 0, 0, 1 + z) :|: z1 = 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(x, 1 + x1, plus(y, times(x1, y)), 1 + z) :|: z2 = y, x1 >= 0, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1, z1 = 1 + x1
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   log(z', z'') -{ 1 }-> loop(1 + x, 1 + (1 + b), 1 + 0, 0) :|: b >= 0, z' = 1 + x, x >= 0, z'' = 1 + (1 + b)
   log(z', z'') -{ 1 }-> 2 :|: b >= 0, z'' = 1 + (1 + b), z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(x'', y), 1 + x'', 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = 1 + x'', z2 = z, y >= 0, z1 = 1 + y, x'' >= 0, z'' = 1 + (1 + b)
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z2 = z, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b), z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, x, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = x, z2 = z, x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b)
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: z' = 1 + (1 + x'), z'' = y, x' >= 0, y >= 0
   times(z', z'') -{ 2 }-> plus(y, 0) :|: z'' = y, y >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(y, 0) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0


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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0


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

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

   { le }
   { plus }
   { times }
   { if, loop }
   { log }

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log}

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 2

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {le}, {plus}, {times}, {if,loop}, {log}
Previous analysis results are:
  le: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 2 }-> if(le(z' - 1, z1 - 1), 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(27) 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''

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(29) 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'

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z' - 1, z'') :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', 0) :|: z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 1 }-> plus(z'', 0) :|: z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 4*z' + 2*z'*z'' + 2*z''

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 1 + (z1 - 1), plus(z2, times(z1 - 1, z2)), 1 + z3) :|: z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 }-> plus(z'', plus(z'', times(z' - 2, z''))) :|: z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {if,loop}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z'']

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {if,loop}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z'']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {if,loop}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z'']
  if: runtime: ?, size: INF
  loop: runtime: ?, size: INF

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z3 :|: z1 >= 0, z3 >= 0, z' = 2, z'' >= 0, z2 >= 0
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(z'', z1, 0, 1 + z3) :|: z1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 2 }-> loop(z'', 0, 0, 1 + z3) :|: z1 = 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   if(z', z'', z1, z2, z3) -{ 5 + 4*z1 + 2*z1*z2 + z2 }-> loop(z'', 1 + (z1 - 1), s7, 1 + z3) :|: s6 >= 0, s6 <= 2 * z2 + 2 * (z2 * (z1 - 1)), s7 >= 0, s7 <= z2 + s6, z1 - 1 >= 0, z3 >= 0, z'' >= 0, z2 >= 0, z' = 1
   le(z', z'') -{ 2 + z'' }-> s :|: s >= 0, s <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   log(z', z'') -{ 1 }-> loop(1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + 0, 0) :|: z'' - 2 >= 0, z' - 1 >= 0
   log(z', z'') -{ 1 }-> 2 :|: z'' - 2 >= 0, z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' >= 0, z'' = 1 + 0
   loop(z', z'', z1, z2) -{ 3 + z1 }-> if(s', 1 + (z' - 1), 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: s' >= 0, s' <= 2, z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' - 1 >= 0
   loop(z', z'', z1, z2) -{ 2 }-> if(2, 0, 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   loop(z', z'', z1, z2) -{ 1 }-> if(0, z', 1 + (1 + (z'' - 2)), 1 + (z1 - 1), z2) :|: z'' - 2 >= 0, z2 >= 0, z' >= 0, z1 - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z' - 1 + z'', z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 3 + z'' }-> s1 :|: s1 >= 0, s1 <= z'' + 0, z'' >= 0, z' = 1 + 0
   times(z', z'') -{ 2 + z'' }-> s2 :|: s2 >= 0, s2 <= z'' + 0, z' - 1 >= 0, z'' >= 0
   times(z', z'') -{ 2 + 4*z' + 2*z'*z'' }-> s5 :|: s3 >= 0, s3 <= 2 * z'' + 2 * (z'' * (z' - 2)), s4 >= 0, s4 <= z'' + s3, s5 >= 0, s5 <= z'' + s4, z' - 2 >= 0, z'' >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {if,loop}, {log}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  times: runtime: O(n^2) [6 + 4*z' + 2*z'*z'' + 2*z''], size: O(n^2) [2*z'*z'' + 2*z'']
  if: runtime: INF, size: INF
  loop: runtime: ?, size: INF

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

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

   le(v0, v1) -> null_le [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   log(v0, v1) -> null_log [0]
   loop(v0, v1, v2, v3) -> null_loop [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

And the following fresh constants:    null_le, null_plus, null_times, null_log, null_loop, null_if

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

(44)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   log(x, 0) -> baseError [1]
   log(x, s(0)) -> baseError [1]
   log(0, s(s(b))) -> logZeroError [1]
   log(s(x), s(s(b))) -> loop(s(x), s(s(b)), s(0), 0) [1]
   loop(x, s(s(b)), s(y), z) -> if(le(x, s(y)), x, s(s(b)), s(y), z) [1]
   if(true, x, b, y, z) -> z [1]
   if(false, x, b, y, z) -> loop(x, b, times(b, y), s(z)) [1]
   le(v0, v1) -> null_le [0]
   plus(v0, v1) -> null_plus [0]
   times(v0, v1) -> null_times [0]
   log(v0, v1) -> null_log [0]
   loop(v0, v1, v2, v3) -> null_loop [0]
   if(v0, v1, v2, v3, v4) -> null_if [0]

The TRS has the following type information:
le :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> false:true:null_le
s :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
0 :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
false :: false:true:null_le
true :: false:true:null_le
plus :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
times :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
log :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
baseError :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
logZeroError :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
loop :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
if :: false:true:null_le -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if -> s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
null_le :: false:true:null_le
null_plus :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
null_times :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
null_log :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
null_loop :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if
null_if :: s:0:baseError:logZeroError:null_plus:null_times:null_log:null_loop:null_if

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

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

   0 => 0
   false => 1
   true => 2
   baseError => 1
   logZeroError => 2
   null_le => 0
   null_plus => 0
   null_times => 0
   null_log => 0
   null_loop => 0
   null_if => 0

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

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

   if(z', z'', z1, z2, z3) -{ 1 }-> z :|: z1 = b, b >= 0, z2 = y, z >= 0, z' = 2, z3 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2, z3) -{ 1 }-> loop(x, b, times(b, y), 1 + z) :|: z1 = b, b >= 0, z2 = y, z >= 0, z3 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2, z3) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, v4 >= 0, z1 = v2, v1 >= 0, z'' = v1, z3 = v4, v2 >= 0, v3 >= 0, z' = v0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   log(z', z'') -{ 1 }-> loop(1 + x, 1 + (1 + b), 1 + 0, 0) :|: b >= 0, z' = 1 + x, x >= 0, z'' = 1 + (1 + b)
   log(z', z'') -{ 1 }-> 2 :|: b >= 0, z'' = 1 + (1 + b), z' = 0
   log(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   log(z', z'') -{ 1 }-> 1 :|: z' = x, x >= 0, z'' = 1 + 0
   log(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   loop(z', z'', z1, z2) -{ 1 }-> if(le(x, 1 + y), x, 1 + (1 + b), 1 + y, z) :|: b >= 0, z >= 0, z' = x, z2 = z, x >= 0, y >= 0, z1 = 1 + y, z'' = 1 + (1 + b)
   loop(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> plus(y, times(x, y)) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   times(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   times(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   log(z0, 0) -> baseError
   log(z0, s(0)) -> baseError
   log(0, s(s(z0))) -> logZeroError
   log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0)
   loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3)
   if(true, z0, z1, z2, z3) -> z3
   if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3))
Tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(0, z0) -> c5
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOG(z0, 0) -> c7
   LOG(z0, s(0)) -> c8
   LOG(0, s(s(z0))) -> c9
   LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(true, z0, z1, z2, z3) -> c12
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
S tuples:
   LE(s(z0), 0) -> c
   LE(0, z0) -> c1
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(0, z0) -> c5
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOG(z0, 0) -> c7
   LOG(z0, s(0)) -> c8
   LOG(0, s(s(z0))) -> c9
   LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(true, z0, z1, z2, z3) -> c12
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
K tuples:none
Defined Rule Symbols:   le_2, plus_2, times_2, log_2, loop_4, if_5

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOG_2, LOOP_4, IF_5

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6_2, c7, c8, c9, c10_1, c11_2, c12, c13_2


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOG(s(z0), s(s(z1))) -> c10(LOOP(s(z0), s(s(z1)), s(0), 0))
Removed 8 trailing nodes:
   LE(0, z0) -> c1
   LOG(z0, 0) -> c7
   IF(true, z0, z1, z2, z3) -> c12
   LOG(z0, s(0)) -> c8
   LOG(0, s(s(z0))) -> c9
   TIMES(0, z0) -> c5
   LE(s(z0), 0) -> c
   PLUS(0, z0) -> c3

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   log(z0, 0) -> baseError
   log(z0, s(0)) -> baseError
   log(0, s(s(z0))) -> logZeroError
   log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0)
   loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3)
   if(true, z0, z1, z2, z3) -> z3
   if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3))
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
K tuples:none
Defined Rule Symbols:   le_2, plus_2, times_2, log_2, loop_4, if_5

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c13_2


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

(51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   log(z0, 0) -> baseError
   log(z0, s(0)) -> baseError
   log(0, s(s(z0))) -> logZeroError
   log(s(z0), s(s(z1))) -> loop(s(z0), s(s(z1)), s(0), 0)
   loop(z0, s(s(z1)), s(z2), z3) -> if(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3)
   if(true, z0, z1, z2, z3) -> z3
   if(false, z0, z1, z2, z3) -> loop(z0, z1, times(z1, z2), s(z3))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2)))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c13_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(z0, s(s(z1)), s(z2), z3) -> c11(IF(le(z0, s(z2)), z0, s(s(z1)), s(z2), z3), LE(z0, s(z2))) by 
   LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2)))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
   LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2)))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
   LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2)))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LOOP(0, s(s(x1)), s(x2), x3) -> c11(IF(true, 0, s(s(x1)), s(x2), x3), LE(0, s(x2)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, z1, z2, z3) -> c13(LOOP(z0, z1, times(z1, z2), s(z3)), TIMES(z1, z2)) by 
   IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
   IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
   IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c13_2


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   IF(false, x0, 0, z0, x3) -> c13(LOOP(x0, 0, 0, s(x3)), TIMES(0, z0))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1)))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c13_2


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(s(z0), s(s(x1)), s(z1), x3) -> c11(IF(le(z0, z1), s(z0), s(s(x1)), s(z1), x3), LE(s(z0), s(z1))) by 
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(IF(true, s(0), s(s(x1)), s(z0), x3), LE(s(0), s(z0)))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2, c11_1


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2, c11_1


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

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
We considered the (Usable) Rules:
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   le(0, z0) -> true
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3 + x_5
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3, x_4)) = x_1 + x_2 + x_4
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = 0
   POL(le(x_1, x_2)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = x_1
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = 0

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2, c11_1


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

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_2
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3, x_4)) = x_1
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1] + x_1 + x_2
   POL(plus(x_1, x_2)) = [1] + x_2
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = [1]

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, IF_5, LOOP_4

Compound Symbols:   c2_1, c4_1, c6_2, c13_2, c11_2, c11_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(z0), z1, x3) -> c13(LOOP(x0, s(z0), plus(z1, times(z0, z1)), s(x3)), TIMES(s(z0), z1)) by 
   IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(0), z0, x3) -> c13(LOOP(x0, s(0), plus(z0, 0), s(x3)), TIMES(s(0), z0))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), 0, x3) -> c13(LOOP(x0, s(x1), times(x1, 0), s(x3)), TIMES(s(x1), 0))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1


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

(75) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   IF(false, x0, s(0), z0, x3) -> c13(TIMES(s(0), z0))
   IF(false, x0, s(x1), 0, x3) -> c(TIMES(s(x1), 0))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1


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

(77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1


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

(79) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
We considered the (Usable) Rules:none
And the Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_3
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3, x_4)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(le(x_1, x_2)) = [1] + x_1 + x_2
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = [1]

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1))))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1


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

(81) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(z0)), s(s(x1)), s(s(z1)), x3) -> c11(IF(le(z0, z1), s(s(z0)), s(s(x1)), s(s(z1)), x3), LE(s(s(z0)), s(s(z1)))) by 
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
S tuples:
   LE(s(z0), s(z1)) -> c2(LE(z0, z1))
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   LE_2, PLUS_2, TIMES_2, LOOP_4, IF_5

Compound Symbols:   c2_1, c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(z0), s(z1)) -> c2(LE(z0, z1)) by 
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2

Compound Symbols:   c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1, c2_1


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

(85) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LOOP(s(0), s(s(x1)), s(z0), x3) -> c11(LE(s(0), s(z0)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3), LE(s(s(z0)), s(0)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
K tuples:
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2

Compound Symbols:   c4_1, c6_2, c11_2, c11_1, c13_2, c13_1, c_1, c2_1


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

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
K tuples:
   LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2)))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, LOOP_4, IF_5, LE_2

Compound Symbols:   c4_1, c6_2, c11_1, c13_2, c13_1, c_1, c11_2, c2_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(x0), s(s(x1)), s(x2), x3) -> c11(LE(s(x0), s(x2))) by 
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0)))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c13_2, c13_1, c_1, c11_2, c2_1, c11_1


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

(91) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, x0, s(x1), s(z0), x3) -> c13(LOOP(x0, s(x1), s(plus(z0, times(x1, s(z0)))), s(x3)), TIMES(s(x1), s(z0))) by 
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c13_2, c13_1, c_1, c11_2, c2_1, c11_1


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

(93) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, x0, s(s(z0)), z1, x3) -> c13(LOOP(x0, s(s(z0)), plus(z1, plus(z1, times(z0, z1))), s(x3)), TIMES(s(s(z0)), z1)) by 
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c13_1, c_1, c11_2, c2_1, c11_1, c13_2


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(s(x2), times(s(x1), s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c13_1, c_1, c11_2, c2_1, c11_1, c13_2


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, x0, s(x1), x2, x3) -> c13(TIMES(s(x1), x2)) by 
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), s(plus(0, times(s(x1), s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(s(x2)), plus(s(s(x2)), times(x1, s(s(x2))))), s(s(x3))), TIMES(s(s(x1)), s(s(x2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1


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

(103) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(LOOP(s(s(x0)), s(s(x1)), plus(s(0), plus(s(0), times(x1, s(0)))), s(x2)), TIMES(s(s(x1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c4(PLUS(z0, z1))
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   PLUS_2, TIMES_2, IF_5, LOOP_4, LE_2

Compound Symbols:   c4_1, c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c4(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_2, c13_1, c4_1


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

(107) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, times(s(z1), s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
----------------------------------------

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1


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

(109) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(times(s(z1), s(0))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1


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

(111) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(z2), plus(s(s(z2)), times(z1, s(s(z2)))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2)))) by IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
----------------------------------------

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c13_2, c4_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(0, plus(s(0), times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2


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

(115) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2


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

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1


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

(119) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
We considered the (Usable) Rules:
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
   le(0, z0) -> true
And the Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(LE(x_1, x_2)) = 0
   POL(LOOP(x_1, x_2, x_3, x_4)) = x_2
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c11(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c13(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = 0
   POL(le(x_1, x_2)) = 0
   POL(plus(x_1, x_2)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = 0

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1


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

(121) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(z0)), s(s(x1)), s(0), x3) -> c11(IF(false, s(s(z0)), s(s(x1)), s(0), x3)) by 
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1


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

(123) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), s(0), x2) -> c13(TIMES(s(s(x1)), s(0))) by 
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(plus(s(0), times(z1, s(0)))), s(z2)), TIMES(s(s(z1)), s(0))) by IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)), TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1


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

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1


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

(129) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c2_1, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1


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

(131) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LE(s(s(y0)), s(s(y1))) -> c2(LE(s(y0), s(y1))) by 
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1, c2_1


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

(133) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) by 
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c1_1, c3_1, c2_1


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

(135) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c1(TIMES(s(s(z1)), s(0))) by 
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c3_1, c2_1, c1_1


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

(137) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(LOOP(s(s(z0)), s(s(z1)), s(s(plus(0, times(z1, s(0))))), s(z2))) by 
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c3_1, c2_1, c1_1


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

(139) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(z0)), s(s(z1)), s(0), z2) -> c3(TIMES(s(s(z1)), s(0))) by 
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c11_1, c13_1, c4_1, c13_2, c2_1, c1_1, c3_1


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

(141) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOP(s(z0), s(s(z1)), s(y1), s(x3)) -> c11(LE(s(z0), s(y1))) by 
   LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1)))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, PLUS_2, LE_2

Compound Symbols:   c6_2, c_1, c11_2, c13_1, c4_1, c13_2, c11_1, c2_1, c1_1, c3_1


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

(143) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c4(PLUS(s(y0), z1)) by 
   PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(s(y0)), z1))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(0, z0) -> 0
   times(s(z0), z1) -> plus(z1, times(z0, z1))
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   le(s(z0), 0) -> false
Tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1)))))
   PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(s(y0)), z1))
S tuples:
   TIMES(s(z0), z1) -> c6(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), s(s(z2)), s(x3)) -> c11(IF(le(z0, z2), s(s(z0)), s(s(z1)), s(s(z2)), s(x3)), LE(s(s(z0)), s(s(z2))))
   IF(false, s(s(z0)), s(s(z1)), s(s(z2)), s(z3)) -> c13(LOOP(s(s(z0)), s(s(z1)), s(s(plus(z2, plus(s(s(z2)), times(z1, s(s(z2))))))), s(s(z3))), TIMES(s(s(z1)), s(s(z2))))
   LE(s(s(s(y0))), s(s(s(y1)))) -> c2(LE(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c4(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, x0, s(x1), 0, x3) -> c(LOOP(x0, s(x1), times(x1, 0), s(x3)))
   IF(false, s(s(x0)), s(s(x1)), s(s(x2)), s(x3)) -> c13(TIMES(s(s(x1)), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c13(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(z0)), s(s(z1)), s(0), s(x2)) -> c11(IF(false, s(s(z0)), s(s(z1)), s(0), s(x2)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c1(TIMES(s(s(x1)), s(0)))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(LOOP(s(s(x0)), s(s(x1)), s(s(plus(0, times(x1, s(0))))), s(s(x2))))
   IF(false, s(s(x0)), s(s(x1)), s(0), s(x2)) -> c3(TIMES(s(s(x1)), s(0)))
   LOOP(s(s(s(y0))), s(s(z1)), s(s(s(y1))), s(z3)) -> c11(LE(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   times_2, plus_2, le_2

Defined Pair Symbols:   TIMES_2, IF_5, LOOP_4, LE_2, PLUS_2

Compound Symbols:   c6_2, c_1, c11_2, c13_1, c13_2, c11_1, c2_1, c1_1, c3_1, c4_1