KILLED
proof of input_OVo0kenmXT.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 3 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 336 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 172 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 267 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 9545 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 6860 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 0 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 657 ms]
    (64) CdtProblem
    (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 17 ms]
    (66) CdtProblem
    (67) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (70) CdtProblem
    (71) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (94) CdtProblem
    (95) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtInstantiationProof [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


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

(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(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   quot(x, s(y)) -> help(x, s(y), 0)
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c)
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y))))
   if(false, x, s(y), c) -> 0

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

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

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


The TRS R consists of the following rules:

   lt(x, 0') -> false
   lt(0', s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0') -> x
   plus(x, s(y)) -> s(plus(x, y))
   quot(x, s(y)) -> help(x, s(y), 0')
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c)
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y))))
   if(false, x, s(y), c) -> 0'

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

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

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


The TRS R consists of the following rules:

   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   plus(x, 0) -> x
   plus(x, s(y)) -> s(plus(x, y))
   quot(x, s(y)) -> help(x, s(y), 0)
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c)
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y))))
   if(false, x, s(y), c) -> 0

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

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

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


The TRS R consists of the following rules:

   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   quot(x, s(y)) -> help(x, s(y), 0) [1]
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1]
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1]
   if(false, x, s(y), c) -> 0 [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(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   quot(x, s(y)) -> help(x, s(y), 0) [1]
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1]
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1]
   if(false, x, s(y), c) -> 0 [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
plus :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
help :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 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:

   quot_2
   help_3
   if_4

(c) The following functions are completely defined:

   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(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   quot(x, s(y)) -> help(x, s(y), 0) [1]
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1]
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1]
   if(false, x, s(y), c) -> 0 [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
plus :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
help :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 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(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   quot(x, s(y)) -> help(x, s(y), 0) [1]
   help(0, s(y), c) -> if(false, 0, s(y), c) [2]
   help(s(y'), s(y), 0) -> if(true, s(y'), s(y), 0) [2]
   help(s(y''), s(y), s(x')) -> if(lt(x', y''), s(y''), s(y), s(x')) [2]
   if(true, x, s(y), c) -> s(help(x, s(y), s(plus(c, y)))) [2]
   if(false, x, s(y), c) -> 0 [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> false:true
0 :: 0:s
false :: false:true
s :: 0:s -> 0:s
true :: false:true
plus :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s
help :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 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
   false => 0
   true => 1

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

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

   help(z, z', z'') -{ 2 }-> if(lt(x', y''), 1 + y'', 1 + y, 1 + x') :|: z' = 1 + y, z = 1 + y'', z'' = 1 + x', y >= 0, x' >= 0, y'' >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + y', 1 + y, 0) :|: z' = 1 + y, z'' = 0, y >= 0, z = 1 + y', y' >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + y, c) :|: z' = 1 + y, c >= 0, y >= 0, z = 0, z'' = c
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z1 = c, c >= 0, x >= 0, y >= 0, z'' = 1 + y, z = 0
   if(z, z', z'', z1) -{ 2 }-> 1 + help(x, 1 + y, 1 + plus(c, y)) :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   quot(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x


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

(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:

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


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

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

   { lt }
   { plus }
   { help, if }
   { quot }

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

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

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

Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot}

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

(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:

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

Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot}

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

(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:

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

Function symbols to be analyzed: {lt}, {plus}, {help,if}, {quot}
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:

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

Function symbols to be analyzed: {plus}, {help,if}, {quot}
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:

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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, z' - 1) :|: z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {plus}, {help,if}, {quot}
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:

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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, z' - 1) :|: z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {plus}, {help,if}, {quot}
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:

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 }-> 1 + help(z', 1 + (z'' - 1), 1 + plus(z1, z'' - 1)) :|: z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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, z' - 1) :|: z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {help,if}, {quot}
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:

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {help,if}, {quot}
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 CoFloCo for: help
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

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

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {help,if}, {quot}
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']
  help: runtime: ?, size: INF
  if: runtime: ?, size: INF

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   help(z, z', z'') -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), 1 + (z'' - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, 1 + (z - 1), 1 + (z' - 1), 0) :|: z'' = 0, z' - 1 >= 0, z - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(0, 0, 1 + (z' - 1), z'') :|: z'' >= 0, z' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z' >= 0, z'' - 1 >= 0, z = 0
   if(z, z', z'', z1) -{ 2 + z'' }-> 1 + help(z', 1 + (z'' - 1), 1 + s1) :|: s1 >= 0, s1 <= z1 + (z'' - 1), z1 >= 0, z = 1, z' >= 0, z'' - 1 >= 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 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   quot(z, z') -{ 1 }-> help(z, 1 + (z' - 1), 0) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {help,if}, {quot}
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']
  help: runtime: INF, size: INF
  if: runtime: ?, size: INF

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

(37) 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:

   quot(v0, v1) -> null_quot [0]
   help(v0, v1, v2) -> null_help [0]
   if(v0, v1, v2, v3) -> null_if [0]
   lt(v0, v1) -> null_lt [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_quot, null_help, null_if, null_lt, null_plus

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

(38)
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(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   quot(x, s(y)) -> help(x, s(y), 0) [1]
   help(x, s(y), c) -> if(lt(c, x), x, s(y), c) [1]
   if(true, x, s(y), c) -> s(help(x, s(y), plus(c, s(y)))) [1]
   if(false, x, s(y), c) -> 0 [1]
   quot(v0, v1) -> null_quot [0]
   help(v0, v1, v2) -> null_help [0]
   if(v0, v1, v2, v3) -> null_if [0]
   lt(v0, v1) -> null_lt [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
lt :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> false:true:null_lt
0 :: 0:s:null_quot:null_help:null_if:null_plus
false :: false:true:null_lt
s :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus
true :: false:true:null_lt
plus :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus
quot :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus
help :: 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus
if :: false:true:null_lt -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus -> 0:s:null_quot:null_help:null_if:null_plus
null_quot :: 0:s:null_quot:null_help:null_if:null_plus
null_help :: 0:s:null_quot:null_help:null_if:null_plus
null_if :: 0:s:null_quot:null_help:null_if:null_plus
null_lt :: false:true:null_lt
null_plus :: 0:s:null_quot:null_help:null_if:null_plus

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

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

   0 => 0
   false => 1
   true => 2
   null_quot => 0
   null_help => 0
   null_if => 0
   null_lt => 0
   null_plus => 0

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

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

   help(z, z', z'') -{ 1 }-> if(lt(c, x), x, 1 + y, c) :|: z' = 1 + y, c >= 0, x >= 0, y >= 0, z = x, z'' = c
   help(z, z', z'') -{ 0 }-> 0 :|: v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z1 = c, c >= 0, z = 1, x >= 0, y >= 0, z'' = 1 + y
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   if(z, z', z'', z1) -{ 1 }-> 1 + help(x, 1 + y, plus(c, 1 + y)) :|: z = 2, z' = x, z1 = c, c >= 0, x >= 0, y >= 0, z'' = 1 + y
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 2 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   quot(z, z') -{ 1 }-> help(x, 1 + y, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   quot(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   quot(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0
Tuples:
   LT(z0, 0) -> c
   LT(0, s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0) -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8
S tuples:
   LT(z0, 0) -> c
   LT(0, s(z0)) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, 0) -> c3
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   IF(false, z0, s(z1), z2) -> c8
K tuples:none
Defined Rule Symbols:   lt_2, plus_2, quot_2, help_3, if_4

Defined Pair Symbols:   LT_2, PLUS_2, QUOT_2, HELP_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5_1, c6_2, c7_2, c8


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   QUOT(z0, s(z1)) -> c5(HELP(z0, s(z1), 0))
Removed 4 trailing nodes:
   IF(false, z0, s(z1), z2) -> c8
   LT(0, s(z0)) -> c1
   PLUS(z0, 0) -> c3
   LT(z0, 0) -> c

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   quot(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2, quot_2, help_3, if_4

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c7_2


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   quot(z0, s(z1)) -> help(z0, s(z1), 0)
   help(z0, s(z1), z2) -> if(lt(z2, z0), z0, s(z1), z2)
   if(true, z0, s(z1), z2) -> s(help(z0, s(z1), plus(z2, s(z1))))
   if(false, z0, s(z1), z2) -> 0

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c7_2


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HELP(z0, s(z1), z2) -> c6(IF(lt(z2, z0), z0, s(z1), z2), LT(z2, z0)) by 
   HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   HELP(0, s(x1), z0) -> c6(IF(false, 0, s(x1), z0), LT(z0, 0))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0), LT(0, s(z0)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2


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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
We considered the (Usable) Rules:
   plus(z0, s(z1)) -> s(plus(z0, z1))
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(HELP(x_1, x_2, x_3)) = [3]x_2 + [2]x_3
   POL(IF(x_1, x_2, x_3, x_4)) = [3]x_3
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(false) = [3]
   POL(lt(x_1, x_2)) = [3] + [3]x_1
   POL(plus(x_1, x_2)) = 0
   POL(s(x_1)) = 0
   POL(true) = [3]

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1)))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
K tuples:
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, z0, s(z1), z2) -> c7(HELP(z0, s(z1), plus(z2, s(z1))), PLUS(z2, s(z1))) by 
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
K tuples:
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c6_1, c7_2


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

(57) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   HELP(s(z0), s(x1), 0) -> c6(IF(true, s(z0), s(x1), 0))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1)))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c7_2


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HELP(s(z1), s(x1), s(z0)) -> c6(IF(lt(z0, z1), s(z1), s(x1), s(z0)), LT(s(z0), s(z1))) by 
   HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(0), s(x1), s(z0)) -> c6(IF(false, s(0), s(x1), s(z0)), LT(s(z0), s(0)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(HELP(x_1, x_2, x_3)) = x_1
   POL(IF(x_1, x_2, x_3, x_4)) = x_2
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(lt(x_1, x_2)) = 0
   POL(plus(x_1, x_2)) = x_1^3 + x_1^2*x_2 + x_1*x_2^2
   POL(s(x_1)) = x_1
   POL(true) = 0

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
K tuples:
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(HELP(x_1, x_2, x_3)) = [1]
   POL(IF(x_1, x_2, x_3, x_4)) = [1]
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1, x_2)) = x_1 + x_2
   POL(false) = [3]
   POL(lt(x_1, x_2)) = [3] + [3]x_1
   POL(plus(x_1, x_2)) = [3]x_1 + [2]x_2
   POL(s(x_1)) = [2]
   POL(true) = [3]

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
K tuples:
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c2_1, c4_1, c7_2, c6_2, c6_1


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

(67) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, x0, s(z1), z0) -> c7(HELP(x0, s(z1), s(plus(z0, z1))), PLUS(z0, s(z1))) by 
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
K tuples:
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c6_1, c7_2


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

(69) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   HELP(s(0), s(x1), s(z0)) -> c6(LT(s(z0), s(0)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
K tuples:
   HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c6_1, c7_2


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

(71) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(s(x0), s(x1), s(x2)) -> c6(LT(s(x2), s(x0))) by 
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
K tuples:
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c2_1, c4_1, c6_2, c7_2, c6_1


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

(73) 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)))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
S tuples:
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)), LT(s(0), s(s(z0))))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, IF_4, LT_2

Compound Symbols:   c4_1, c6_2, c7_2, c6_1, c2_1


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
S tuples:
   PLUS(z0, s(z1)) -> c4(PLUS(z0, z1))
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
K tuples:
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, IF_4, LT_2

Compound Symbols:   c4_1, c6_2, c7_2, c6_1, c2_1


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

(77) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(z1)) -> c4(PLUS(z0, z1)) by 
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
K tuples:
   HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c6_2, c7_2, c6_1, c2_1, c4_1


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

(79) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HELP(s(s(x0)), s(x1), s(y0)) -> c6(LT(s(y0), s(s(x0)))) by 
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1)))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c6_2, c7_2, c2_1, c6_1, c4_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, s(s(x0)), s(x1), s(0)) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(0), x1))), PLUS(s(0), s(x1))) by 
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0)))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))), PLUS(s(0), s(0)))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c6_2, c7_2, c2_1, c6_1, c4_1


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c6_2, c7_2, c2_1, c6_1, c4_1, c7_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, s(s(x0)), s(x1), s(s(x2))) -> c7(HELP(s(s(x0)), s(x1), s(plus(s(s(x2)), x1))), PLUS(s(s(x2)), s(x1))) by 
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0)))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0)))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c6_1, c4_1, c7_2, c7_1


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

(87) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   HELP(s(s(z0)), s(x1), s(0)) -> c6(IF(true, s(s(z0)), s(x1), s(0)))
   IF(true, s(s(x0)), s(0), s(0)) -> c7(HELP(s(s(x0)), s(0), s(s(0))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0)))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))), PLUS(s(s(x2)), s(0)))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c4_1, c6_1, c7_2


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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))), PLUS(s(0), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c4_1, c6_1, c7_2, c7_1


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1


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

(93) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(PLUS(s(0), s(s(z1))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1


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

(95) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, LT_2, PLUS_2, IF_4

Compound Symbols:   c6_2, c2_1, c4_1, c6_1, c7_2, c7_1, c_1


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

(97) 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))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
S tuples:
   HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   HELP_3, PLUS_2, IF_4, LT_2

Compound Symbols:   c6_2, c4_1, c6_1, c7_2, c7_1, c_1, c2_1


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

(99) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(s(s(z1)), s(x1), s(s(z0))) -> c6(IF(lt(z0, z1), s(s(z1)), s(x1), s(s(z0))), LT(s(s(z0)), s(s(z1)))) by 
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
S tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
K tuples:
   HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, IF_4, LT_2

Compound Symbols:   c4_1, c6_1, c7_2, c7_1, c_1, c2_1, c6_2


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(s(s(z0)), s(z1), s(s(y0))) -> c6(LT(s(s(y0)), s(s(z0)))) by 
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
S tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, LT_2, HELP_3

Compound Symbols:   c4_1, c7_2, c7_1, c_1, c2_1, c6_2, c6_1


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

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, s(s(x0)), s(0), s(s(x2))) -> c7(HELP(s(s(x0)), s(0), s(s(s(x2))))) by 
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
S tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, LT_2, HELP_3

Compound Symbols:   c4_1, c7_2, c_1, c2_1, c6_2, c6_1, c7_1


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

(105) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(0), s(s(s(x1)))), LT(s(s(s(x1))), s(s(x0)))) by 
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
S tuples:
   PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1)))
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, LT_2, HELP_3

Compound Symbols:   c4_1, c7_2, c_1, c2_1, c6_2, c6_1, c7_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c4(PLUS(z0, s(y1))) by 
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
S tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_4, LT_2, HELP_3, PLUS_2

Compound Symbols:   c7_2, c_1, c2_1, c6_2, c6_1, c7_1, c4_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(LT(s(s(y0)), s(s(x0)))) by 
   HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1)))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
   HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0))))
   HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_4, LT_2, HELP_3, PLUS_2

Compound Symbols:   c7_2, c_1, c2_1, c6_2, c6_1, c7_1, c4_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HELP(s(s(x0)), s(0), s(s(s(x1)))) -> c6(LT(s(s(s(x1))), s(s(x0)))) by 
   HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1)))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
Tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
   HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1)))))
   HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1)))))
S tuples:
   IF(true, s(s(x0)), s(s(z1)), s(s(x2))) -> c7(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(s(x2)), z1)))), PLUS(s(s(x2)), s(s(z1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c2(LT(s(s(y0)), s(s(y1))))
   HELP(s(s(x0)), s(s(x1)), s(s(y0))) -> c6(IF(lt(y0, x0), s(s(x0)), s(s(x1)), s(s(y0))), LT(s(s(y0)), s(s(x0))))
   IF(true, s(s(x0)), s(0), s(s(s(x1)))) -> c7(HELP(s(s(x0)), s(0), s(s(s(s(x1))))))
   HELP(s(s(x0)), s(0), s(s(s(s(x1))))) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(0), s(s(s(s(x1))))), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(z0, s(s(s(y1)))) -> c4(PLUS(z0, s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(s(z1)), s(0)) -> c(HELP(s(s(x0)), s(s(z1)), s(s(plus(s(0), z1)))))
   HELP(s(s(s(y1))), s(s(z1)), s(s(s(y0)))) -> c6(LT(s(s(s(y0))), s(s(s(y1)))))
   HELP(s(s(s(y1))), s(0), s(s(s(z1)))) -> c6(LT(s(s(s(z1))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_4, LT_2, HELP_3, PLUS_2

Compound Symbols:   c7_2, c_1, c2_1, c6_2, c7_1, c4_1, c6_1