KILLED
proof of input_R2RwzUVYVD.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 362 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 127 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 262 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 447 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 158 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 4322 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 438 ms]
        (42) CpxRNTS
    (43) CompletionProof [UPPER BOUND(ID), 0 ms]
    (44) CpxTypedWeightedCompleteTrs
        (45) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (46) CpxRNTS
(47) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(48) CdtProblem
    (49) CdtLeafRemovalProof [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 [ComplexityIfPolyImplication, 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 2 ms]
    (62) CdtProblem
    (63) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (66) CdtProblem
    (67) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (70) CdtProblem
    (71) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (94) CdtProblem
    (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 25 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 7012 ms]
    (118) CdtProblem
    (119) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) 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:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0), s(0))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> y

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

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

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


The TRS R consists of the following rules:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0', y) -> 0'
   times(s(x), y) -> plus(y, times(x, y))
   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0'), s(0'))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> y

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

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

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


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   times(0, y) -> 0
   times(s(x), y) -> plus(y, times(x, y))
   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   fac(x) -> loop(x, s(0), s(0))
   loop(x, c, y) -> if(lt(x, c), x, c, y)
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c)))
   if(true, x, c, y) -> y

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

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

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


The TRS R consists of the following rules:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   fac(x) -> loop(x, s(0), s(0)) [1]
   loop(x, c, y) -> if(lt(x, c), x, c, y) [1]
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c))) [1]
   if(true, x, c, y) -> y [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:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   fac(x) -> loop(x, s(0), s(0)) [1]
   loop(x, c, y) -> if(lt(x, c), x, c, y) [1]
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c))) [1]
   if(true, x, c, y) -> y [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
times :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
fac :: 0:s -> 0:s
loop :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s

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

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

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

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

   fac_1
   loop_3
   if_4

(c) The following functions are completely defined:

   times_2
   lt_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   fac(x) -> loop(x, s(0), s(0)) [1]
   loop(x, c, y) -> if(lt(x, c), x, c, y) [1]
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c))) [1]
   if(true, x, c, y) -> y [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
times :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
fac :: 0:s -> 0:s
loop :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s

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

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

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

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(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]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   fac(x) -> loop(x, s(0), s(0)) [1]
   loop(0, s(x''), y) -> if(true, 0, s(x''), y) [2]
   loop(x, 0, y) -> if(false, x, 0, y) [2]
   loop(s(x1), s(y'), y) -> if(lt(x1, y'), s(x1), s(y'), y) [2]
   if(false, x, c, 0) -> loop(x, s(c), 0) [2]
   if(false, x, c, s(x2)) -> loop(x, s(c), plus(s(c), times(x2, s(c)))) [2]
   if(true, x, c, y) -> y [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
times :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
fac :: 0:s -> 0:s
loop :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s

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

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

   0 => 0
   true => 1
   false => 0

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

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

   fac(z) -{ 1 }-> loop(x, 1 + 0, 1 + 0) :|: x >= 0, z = x
   if(z, z', z'', z1) -{ 1 }-> y :|: z1 = y, z' = x, c >= 0, z = 1, x >= 0, y >= 0, z'' = c
   if(z, z', z'', z1) -{ 2 }-> loop(x, 1 + c, plus(1 + c, times(x2, 1 + c))) :|: z' = x, c >= 0, x >= 0, z = 0, z'' = c, x2 >= 0, z1 = 1 + x2
   if(z, z', z'', z1) -{ 2 }-> loop(x, 1 + c, 0) :|: z1 = 0, z' = x, c >= 0, x >= 0, z = 0, z'' = c
   loop(z, z', z'') -{ 2 }-> if(lt(x1, y'), 1 + x1, 1 + y', y) :|: x1 >= 0, z'' = y, y >= 0, z = 1 + x1, y' >= 0, z' = 1 + y'
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + x'', y) :|: z' = 1 + x'', z'' = y, y >= 0, x'' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, x, 0, y) :|: z'' = y, x >= 0, y >= 0, z = x, z' = 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 2 }-> plus(y, plus(y, times(x', y))) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y
   times(z, z') -{ 2 }-> plus(y, 0) :|: z = 1 + 0, y >= 0, z' = y
   times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y


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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: 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 = 1 + 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


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

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

   { lt }
   { plus }
   { times }
   { if, loop }
   { fac }

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: 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 = 1 + 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {lt}, {plus}, {times}, {if,loop}, {fac}

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: 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 = 1 + 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {lt}, {plus}, {times}, {if,loop}, {fac}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: 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 = 1 + 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: lt
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:

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(lt(z - 1, z' - 1), 1 + (z - 1), 1 + (z' - 1), z'') :|: z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: 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 = 1 + 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

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

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

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

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

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

(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:

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

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', plus(1 + z'', times(z1 - 1, 1 + z''))) :|: z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= 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

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

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

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

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 3 + 2*z''*z1 + 6*z1 }-> loop(z', 1 + z'', s6) :|: s5 >= 0, s5 <= 1 + z'' + 2 * ((1 + z'') * (z1 - 1)), s6 >= 0, s6 <= 1 + z'' + s5, z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= z' + 0, z = 1 + 0, z' >= 0
   times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s4 :|: s2 >= 0, s2 <= z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= z' + s2, s4 >= 0, s4 <= z' + s3, z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: 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:

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 3 + 2*z''*z1 + 6*z1 }-> loop(z', 1 + z'', s6) :|: s5 >= 0, s5 <= 1 + z'' + 2 * ((1 + z'') * (z1 - 1)), s6 >= 0, s6 <= 1 + z'' + s5, z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= z' + 0, z = 1 + 0, z' >= 0
   times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s4 :|: s2 >= 0, s2 <= z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= z' + s2, s4 >= 0, s4 <= z' + s3, z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   fac(z) -{ 1 }-> loop(z, 1 + 0, 1 + 0) :|: z >= 0
   if(z, z', z'', z1) -{ 1 }-> z1 :|: z'' >= 0, z = 1, z' >= 0, z1 >= 0
   if(z, z', z'', z1) -{ 3 + 2*z''*z1 + 6*z1 }-> loop(z', 1 + z'', s6) :|: s5 >= 0, s5 <= 1 + z'' + 2 * ((1 + z'') * (z1 - 1)), s6 >= 0, s6 <= 1 + z'' + s5, z'' >= 0, z' >= 0, z = 0, z1 - 1 >= 0
   if(z, z', z'', z1) -{ 2 }-> loop(z', 1 + z'', 0) :|: z1 = 0, z'' >= 0, z' >= 0, z = 0
   loop(z, z', z'') -{ 3 + z' }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'') :|: s' >= 0, s' <= 1, z - 1 >= 0, z'' >= 0, z' - 1 >= 0
   loop(z, z', z'') -{ 2 }-> if(1, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   loop(z, z', z'') -{ 2 }-> if(0, z, 0, z'') :|: z >= 0, z'' >= 0, z' = 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s1 :|: s1 >= 0, s1 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> s'' :|: s'' >= 0, s'' <= z' + 0, z = 1 + 0, z' >= 0
   times(z, z') -{ 4*z + 2*z*z' + -1*z' }-> s4 :|: s2 >= 0, s2 <= z' + 2 * (z' * (z - 2)), s3 >= 0, s3 <= z' + s2, s4 >= 0, s4 <= z' + s3, z - 2 >= 0, z' >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {if,loop}, {fac}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: O(n^2) [4 + 4*z + 2*z*z' + z'], size: O(n^2) [2*z*z' + 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:
none

And the following fresh constants: none

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

(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:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(y, times(x, y)) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   fac(x) -> loop(x, s(0), s(0)) [1]
   loop(x, c, y) -> if(lt(x, c), x, c, y) [1]
   if(false, x, c, y) -> loop(x, s(c), times(y, s(c))) [1]
   if(true, x, c, y) -> y [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
times :: 0:s -> 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
fac :: 0:s -> 0:s
loop :: 0:s -> 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s

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
   true => 1
   false => 0

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

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

   fac(z) -{ 1 }-> loop(x, 1 + 0, 1 + 0) :|: x >= 0, z = x
   if(z, z', z'', z1) -{ 1 }-> y :|: z1 = y, z' = x, c >= 0, z = 1, x >= 0, y >= 0, z'' = c
   if(z, z', z'', z1) -{ 1 }-> loop(x, 1 + c, times(y, 1 + c)) :|: z1 = y, z' = x, c >= 0, x >= 0, y >= 0, z = 0, z'' = c
   loop(z, z', z'') -{ 1 }-> if(lt(x, c), x, c, y) :|: z'' = y, c >= 0, x >= 0, y >= 0, z' = c, z = x
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 1 }-> plus(y, times(x, y)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, z' = y

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:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   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))
   fac(z0) -> loop(z0, s(0), s(0))
   loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
   if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
   if(true, z0, z1, z2) -> z2
Tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0, z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0, z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10
S tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(0, z0) -> c3
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(0, z0) -> c5
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   IF(true, z0, z1, z2) -> c10
K tuples:none
Defined Rule Symbols:   lt_2, times_2, plus_2, fac_1, loop_3, if_4

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, FAC_1, LOOP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_2, c5, c6_1, c7_1, c8_2, c9_2, c10


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FAC(z0) -> c7(LOOP(z0, s(0), s(0)))
Removed 5 trailing nodes:
   IF(true, z0, z1, z2) -> c10
   LT(z0, 0) -> c1
   PLUS(0, z0) -> c5
   LT(0, s(z0)) -> c
   TIMES(0, z0) -> c3

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   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))
   fac(z0) -> loop(z0, s(0), s(0))
   loop(z0, z1, z2) -> if(lt(z0, z1), z0, z1, z2)
   if(false, z0, z1, z2) -> loop(z0, s(z1), times(z2, s(z1)))
   if(true, z0, z1, z2) -> z2
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, times_2, plus_2, fac_1, loop_3, if_4

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c9_2


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

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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c9_2


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(z0, z1, z2) -> c8(IF(lt(z0, z1), z0, z1, z2), LT(z0, z1)) by 
   LOOP(0, s(z0), x2) -> c8(IF(true, 0, s(z0), x2), LT(0, s(z0)))
   LOOP(z0, 0, x2) -> c8(IF(false, z0, 0, x2), LT(z0, 0))
   LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(0, s(z0), x2) -> c8(IF(true, 0, s(z0), x2), LT(0, s(z0)))
   LOOP(z0, 0, x2) -> c8(IF(false, z0, 0, x2), LT(z0, 0))
   LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(s(z0), s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(0, s(z0), x2) -> c8(IF(true, 0, s(z0), x2), LT(0, s(z0)))
   LOOP(z0, 0, x2) -> c8(IF(false, z0, 0, x2), LT(z0, 0))
   LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c8_2


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

(55) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOOP(z0, 0, x2) -> c8(IF(false, z0, 0, x2), LT(z0, 0))
Removed 1 trailing nodes:
   LOOP(0, s(z0), x2) -> c8(IF(true, 0, s(z0), x2), LT(0, s(z0)))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(s(z0), s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c8_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace LOOP(s(z0), s(z1), x2) -> c8(IF(lt(z0, z1), s(z0), s(z1), x2), LT(s(z0), s(z1))) by 
   LOOP(s(0), s(s(z0)), x2) -> c8(IF(true, s(0), s(s(z0)), x2), LT(s(0), s(s(z0))))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(IF(true, s(0), s(s(z0)), x2), LT(s(0), s(s(z0))))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(IF(true, s(0), s(s(z0)), x2), LT(s(0), s(s(z0))))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c8_2, c8_1


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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c8_2, c8_1


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

(61) 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(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(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)) = [1] + x_1
   POL(LOOP(x_1, x_2, x_3)) = x_2
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c2(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(false) = 0
   POL(lt(x_1, x_2)) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = 0

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1)))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, IF_4, LOOP_3

Compound Symbols:   c2_1, c4_2, c6_1, c9_2, c8_2, c8_1


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

(63) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, z0, z1, z2) -> c9(LOOP(z0, s(z1), times(z2, s(z1))), TIMES(z2, s(z1))) by 
   IF(false, s(x0), s(0), x1) -> c9(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))), TIMES(x1, s(s(0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c9(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))), TIMES(x1, s(s(0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(0)))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   IF(false, s(x0), s(0), x1) -> c9(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))), TIMES(x1, s(s(0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c8_1, c9_2


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

(65) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   LOOP(s(z0), s(0), x2) -> c8(IF(false, s(z0), s(0), x2), LT(s(z0), s(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c9(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))), TIMES(x1, s(s(0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   IF(false, s(x0), s(0), x1) -> c9(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))), TIMES(x1, s(s(0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c8_1, c9_2


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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(x0), s(0), x1) -> c(TIMES(x1, s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(x0), s(0), x1) -> c(TIMES(x1, s(s(0))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c8_1, c9_2, c_1


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

(69) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF(false, s(x0), s(0), x1) -> c(TIMES(x1, s(s(0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c8_1, c9_2, c_1


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

(71) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_2, c8_1, c9_2, c_1


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

(73) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(z0)), s(s(z1)), x2) -> c8(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x2), LT(s(s(z0)), s(s(z1)))) by 
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c8(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0), LT(s(s(z0)), s(s(0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c8(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0), LT(s(s(z0)), s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c8(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0), LT(s(s(z0)), s(s(0))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_1, c9_2, c_1, c8_2


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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_1, c9_2, c_1, c8_2, c1_1


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

(77) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_1, c9_2, c_1, c8_2, c1_1


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0)) by LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
----------------------------------------

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   LT_2, TIMES_2, PLUS_2, LOOP_3, IF_4

Compound Symbols:   c2_1, c4_2, c6_1, c8_1, c9_2, c_1, c8_2, c1_1


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

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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(IF(lt(z0, 0), s(s(z0)), s(s(0)), y0))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, PLUS_2, LOOP_3, IF_4, LT_2

Compound Symbols:   c4_2, c6_1, c8_1, c9_2, c_1, c8_2, c1_1, c2_1


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   LOOP(s(0), s(s(z0)), x2) -> c8(LT(s(0), s(s(z0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1)))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, PLUS_2, LOOP_3, IF_4, LT_2

Compound Symbols:   c4_2, c6_1, c8_1, c9_2, c_1, c8_2, c1_1, c2_1


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

(85) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(x0), s(x1), x2) -> c8(LT(s(x0), s(x1))) by 
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(z0), z1) -> c6(PLUS(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, PLUS_2, IF_4, LOOP_3, LT_2

Compound Symbols:   c4_2, c6_1, c9_2, c_1, c8_2, c1_1, c2_1, c8_1


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

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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c9_2, c_1, c8_2, c1_1, c2_1, c8_1, c6_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(x0)), s(s(x1)), x2) -> c9(LOOP(s(s(x0)), s(s(s(x1))), times(x2, s(s(s(x1))))), TIMES(x2, s(s(s(x1))))) by 
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c9(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))), TIMES(x1, s(s(s(0)))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c9(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))), TIMES(x1, s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c9(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))), TIMES(x1, s(s(s(0)))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c8_2, c1_1, c2_1, c8_1, c6_1, c9_2


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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c8_2, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1


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

(93) 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(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(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_1
   POL(LOOP(x_1, x_2, x_3)) = x_2
   POL(LT(x_1, x_2)) = 0
   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(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(lt(x_1, x_2)) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = [1]

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1)))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c8_2, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1


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

(95) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(IF(lt(x0, s(x1)), s(s(x0)), s(s(s(x1))), y0), LT(s(s(x0)), s(s(s(x1))))) by 
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0), LT(s(s(x0)), s(s(s(0)))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0), LT(s(s(x0)), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0), LT(s(s(x0)), s(s(s(0)))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1, c8_2


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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1, c8_2, c5_1


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

(99) 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(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(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_1 + x_2
   POL(LOOP(x_1, x_2, x_3)) = [1] + x_1
   POL(LT(x_1, x_2)) = 0
   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(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(c9(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(lt(x_1, x_2)) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]
   POL(times(x_1, x_2)) = [1] + x_1 + x_2
   POL(true) = [1]

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1)))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1, c8_2, c5_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace LOOP(s(s(x0)), s(s(s(x1))), y0) -> c8(LT(s(s(x0)), s(s(s(x1))))) by 
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c2_1, c8_1, c6_1, c9_2, c3_1, c8_2, c5_1


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

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

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, PLUS_2, LT_2

Compound Symbols:   c4_2, c_1, c1_1, c8_1, c6_1, c9_2, c3_1, c8_2, c5_1, c2_1


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

(105) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   LOOP(s(s(z0)), s(s(0)), y0) -> c1(LT(s(s(z0)), s(s(0))))
   LOOP(s(x0), s(s(0)), y0) -> c8(LT(s(x0), s(s(0))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, PLUS_2, LT_2

Compound Symbols:   c4_2, c_1, c1_1, c6_1, c9_2, c3_1, c8_2, c5_1, c8_1, c2_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c6(PLUS(s(y0), z1)) by 
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c8_1, c2_1, c6_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(false, s(s(x0)), s(s(0)), x1) -> c3(TIMES(x1, s(s(s(0))))) by 
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c8_1, c2_1, c6_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(LT(s(s(x0)), s(s(s(0))))) by 
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c8_1, c2_1, c6_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(LT(s(s(x0)), s(s(s(s(x1)))))) by 
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c8_1, c2_1, c6_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LOOP(s(s(x0)), s(s(s(0))), y0) -> c8(LT(s(s(x0)), s(s(s(0))))) by 
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c2_1, c6_1, c8_1


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

(117) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
We considered the (Usable) Rules:none
And the Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
The order we found is given by the following interpretation:

Matrix interpretation [MATRO]:

Non-tuple symbols: 
<<<
 M( s_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[0, 1, 3], [0, 0, 1], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( true ) =  	[[2], [1], [2]]
>>>

<<<
 M( plus_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( 0 ) =  	[[0], [2], [2]]
>>>

<<<
 M( lt_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( times_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( false ) =  	[[0], [0], [0]]
>>>

Tuple symbols: 
<<<
 M( c_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c1_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( TIMES_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( LOOP_3(x_1, ..., x_3) ) =  	[[0], [2], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_3
>>>

<<<
 M( c8_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( c8_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c5_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c6_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( IF_4(x_1, ..., x_4) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_3 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_4
>>>

<<<
 M( c9_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( c2_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( LT_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( PLUS_2(x_1, x_2) ) =  	[[0], [0], [1]] 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[0, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>

<<<
 M( c3_1(x_1) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1
>>>

<<<
 M( c4_2(x_1, x_2) ) =  	[[0], [0], [0]] 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_1 	+ 	[[1, 0, 0], [0, 0, 0], [0, 0, 0]] 	* 	x_2
>>>



Matrix type: 

We used a basic matrix type which is not further parametrizeable.





As matrix orders are CE-compatible, we used usable rules w.r.t. argument filtering in the order.
----------------------------------------

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c2_1, c6_1, c8_1


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

(119) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))

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

(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))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
Tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   LOOP(s(s(z0)), s(s(0)), z1) -> c1(IF(false, s(s(z0)), s(s(0)), z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
S tuples:
   TIMES(s(z0), z1) -> c4(PLUS(z1, times(z0, z1)), TIMES(z0, z1))
   IF(false, s(s(x0)), s(s(s(x1))), x2) -> c9(LOOP(s(s(x0)), s(s(s(s(x1)))), times(x2, s(s(s(s(x1)))))), TIMES(x2, s(s(s(s(x1))))))
   LOOP(s(s(x0)), s(s(s(s(x1)))), y0) -> c8(IF(lt(x0, s(s(x1))), s(s(x0)), s(s(s(s(x1)))), y0), LT(s(s(x0)), s(s(s(s(x1))))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c6(PLUS(s(s(y0)), z1))
K tuples:
   IF(false, s(x0), s(0), x1) -> c(LOOP(s(x0), s(s(0)), times(x1, s(s(0)))))
   IF(false, s(s(z0)), s(s(0)), s(y0)) -> c3(TIMES(s(y0), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c5(LT(s(s(s(y0))), s(s(s(0)))))
   LOOP(s(s(s(y0))), s(s(s(s(z1)))), z2) -> c8(LT(s(s(s(y0))), s(s(s(s(z1))))))
   LOOP(s(s(s(y0))), s(s(s(0))), z1) -> c8(LT(s(s(s(y0))), s(s(s(0)))))
   IF(false, s(s(x0)), s(s(0)), x1) -> c3(LOOP(s(s(x0)), s(s(s(0))), times(x1, s(s(s(0))))))
   LOOP(s(s(x0)), s(s(s(0))), y0) -> c5(IF(lt(x0, s(0)), s(s(x0)), s(s(s(0))), y0))
Defined Rule Symbols:   times_2, plus_2, lt_2

Defined Pair Symbols:   TIMES_2, IF_4, LOOP_3, LT_2, PLUS_2

Compound Symbols:   c4_2, c_1, c1_1, c9_2, c3_1, c8_2, c5_1, c2_1, c6_1, c8_1