KILLED
proof of input_ajhKP9HHZm.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 359 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 144 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 480 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 1328 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 258 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 [ComplexityIfPolyImplication, 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 29 ms]
    (58) CdtProblem
    (59) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (74) CdtProblem
    (75) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (78) CdtProblem
    (79) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [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), 7 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) 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:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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


The TRS R consists of the following rules:

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

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

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

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

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

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

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:

   times_2

(c) The following functions are completely defined:

   lt_2
   help_3
   if_4
   plus_2

Due to the following rules being added:

   lt(v0, v1) -> null_lt [0]
   if(v0, v1, v2, v3) -> 0 [0]

And the following fresh constants:    null_lt

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

(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:

   times(x, y) -> help(x, y, 0) [1]
   help(x, y, c) -> if(lt(c, y), x, y, c) [1]
   if(true, x, y, c) -> plus(x, help(x, y, s(c))) [1]
   if(false, x, y, c) -> 0 [1]
   lt(0, s(x)) -> true [1]
   lt(s(x), 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(v0, v1) -> null_lt [0]
   if(v0, v1, v2, v3) -> 0 [0]

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

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:

   times(x, y) -> help(x, y, 0) [1]
   help(x, s(x'), 0) -> if(true, x, s(x'), 0) [2]
   help(x, 0, s(x'')) -> if(false, x, 0, s(x'')) [2]
   help(x, s(y'), s(x1)) -> if(lt(x1, y'), x, s(y'), s(x1)) [2]
   help(x, y, c) -> if(null_lt, x, y, c) [1]
   if(true, x, y, c) -> plus(x, if(lt(s(c), y), x, y, s(c))) [2]
   if(false, x, y, c) -> 0 [1]
   lt(0, s(x)) -> true [1]
   lt(s(x), 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   plus(x, 0) -> x [1]
   plus(0, x) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(v0, v1) -> null_lt [0]
   if(v0, v1, v2, v3) -> 0 [0]

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

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

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

   0 => 0
   true => 2
   false => 1
   null_lt => 0

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

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

   help(z, z', z'') -{ 2 }-> if(lt(x1, y'), x, 1 + y', 1 + x1) :|: x1 >= 0, z'' = 1 + x1, x >= 0, y' >= 0, z' = 1 + y', z = x
   help(z, z', z'') -{ 2 }-> if(2, x, 1 + x', 0) :|: z'' = 0, z' = 1 + x', x >= 0, x' >= 0, z = x
   help(z, z', z'') -{ 2 }-> if(1, x, 0, 1 + x'') :|: x >= 0, z'' = 1 + x'', x'' >= 0, z = x, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, x, y, c) :|: c >= 0, x >= 0, y >= 0, z = x, z' = y, z'' = c
   if(z, z', z'', z1) -{ 2 }-> plus(x, if(lt(1 + c, y), x, y, 1 + c)) :|: z = 2, z' = x, z'' = y, z1 = c, c >= 0, x >= 0, y >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z' = x, z'' = y, z1 = c, c >= 0, z = 1, x >= 0, y >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, v2 >= 0, v3 >= 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = 1 + 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') -{ 1 }-> x :|: z' = x, x >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 1 }-> help(x, y, 0) :|: x >= 0, y >= 0, z = x, z' = y


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

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

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

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


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

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

   { lt }
   { plus }
   { if }
   { help }
   { times }

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

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

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

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

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

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

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

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

(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: 2

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

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

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

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

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

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

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

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

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

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

   help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: 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
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: 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
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: 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
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

Function symbols to be analyzed: {if}, {help}, {times}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z'], size: O(1) [2]
  plus: runtime: O(n^1) [1 + z + 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, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   help(z, z', z'') -{ 3 + z' }-> if(s, z, 1 + (z' - 1), 1 + (z'' - 1)) :|: s >= 0, s <= 2, z'' - 1 >= 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(2, z, 1 + (z' - 1), 0) :|: z'' = 0, z >= 0, z' - 1 >= 0
   help(z, z', z'') -{ 2 }-> if(1, z, 0, 1 + (z'' - 1)) :|: z >= 0, z'' - 1 >= 0, z' = 0
   help(z, z', z'') -{ 1 }-> if(0, z, z', z'') :|: z'' >= 0, z >= 0, z' >= 0
   if(z, z', z'', z1) -{ 4 + z'' }-> plus(z', if(s', z', z'', 1 + z1)) :|: s' >= 0, s' <= 2, z = 2, z1 >= 0, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 1 }-> 0 :|: z1 >= 0, z = 1, z' >= 0, z'' >= 0
   if(z, z', z'', z1) -{ 0 }-> 0 :|: z >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 2 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: z - 1 >= 0, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s1 :|: s1 >= 0, s1 <= z + (z' - 1), z >= 0, z' - 1 >= 0
   plus(z, z') -{ 1 + z + z' }-> 1 + s2 :|: s2 >= 0, s2 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 1 }-> help(z, z', 0) :|: z >= 0, z' >= 0

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

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

And the following fresh constants:    null_lt, null_if, 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:

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

The TRS has the following type information:
times :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus
help :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus
0 :: 0:s:null_if:null_plus
if :: true:false:null_lt -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus
lt :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> true:false:null_lt
true :: true:false:null_lt
plus :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus -> 0:s:null_if:null_plus
s :: 0:s:null_if:null_plus -> 0:s:null_if:null_plus
false :: true:false:null_lt
null_lt :: true:false:null_lt
null_if :: 0:s:null_if:null_plus
null_plus :: 0:s: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
   true => 2
   false => 1
   null_lt => 0
   null_if => 0
   null_plus => 0

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

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

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

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:
   times(z0, z1) -> help(z0, z1, 0)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   TIMES(z0, z1) -> c(HELP(z0, z1, 0))
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   IF(false, z0, z1, z2) -> c3
   LT(0, s(z0)) -> c4
   LT(s(z0), 0) -> c5
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, 0) -> c7
   PLUS(0, z0) -> c8
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
S tuples:
   TIMES(z0, z1) -> c(HELP(z0, z1, 0))
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   IF(false, z0, z1, z2) -> c3
   LT(0, s(z0)) -> c4
   LT(s(z0), 0) -> c5
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, 0) -> c7
   PLUS(0, z0) -> c8
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, help_3, if_4, lt_2, plus_2

Defined Pair Symbols:   TIMES_2, HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c_1, c1_2, c2_2, c3, c4, c5, c6_1, c7, c8, c9_1, c10_1


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TIMES(z0, z1) -> c(HELP(z0, z1, 0))
Removed 5 trailing nodes:
   PLUS(z0, 0) -> c7
   PLUS(0, z0) -> c8
   LT(0, s(z0)) -> c4
   LT(s(z0), 0) -> c5
   IF(false, z0, z1, z2) -> c3

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> help(z0, z1, 0)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
S tuples:
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, help_3, if_4, lt_2, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c6_1, c9_1, c10_1


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   times(z0, z1) -> help(z0, z1, 0)

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
S tuples:
   HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1))
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   HELP_3, IF_4, LT_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c6_1, c9_1, c10_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HELP(z0, z1, z2) -> c1(IF(lt(z2, z1), z0, z1, z2), LT(z2, z1)) by 
   HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0)))
   HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0)))
   HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0)))
   HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, LT_2, PLUS_2, HELP_3

Compound Symbols:   c2_2, c6_1, c9_1, c10_1, c1_2


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   HELP(x0, s(z0), 0) -> c1(IF(true, x0, s(z0), 0), LT(0, s(z0)))
Removed 1 trailing nodes:
   HELP(x0, 0, s(z0)) -> c1(IF(false, x0, 0, s(z0)), LT(s(z0), 0))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
S tuples:
   IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2)))
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, LT_2, PLUS_2, HELP_3

Compound Symbols:   c2_2, c6_1, c9_1, c10_1, c1_2


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, z0, z1, z2) -> c2(PLUS(z0, help(z0, z1, s(z2))), HELP(z0, z1, s(z2))) by 
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1)))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, HELP_3, IF_4

Compound Symbols:   c6_1, c9_1, c10_1, c1_2, c2_2, c2_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace HELP(x0, s(z1), s(z0)) -> c1(IF(lt(z0, z1), x0, s(z1), s(z0)), LT(s(z0), s(z1))) by 
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(0), s(s(z0))) -> c1(IF(false, x0, s(0), s(s(z0))), LT(s(s(z0)), s(0)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1


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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
K tuples:none
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1


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

(57) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
We considered the (Usable) Rules:
   lt(s(z0), 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
And the Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(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] + x_2 + x_3
   POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(false) = [1]
   POL(help(x_1, x_2, x_3)) = [1] + x_2 + x_3
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(lt(x_1, x_2)) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c6(LT(z0, z1))
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   LT_2, PLUS_2, IF_4, HELP_3

Compound Symbols:   c6_1, c9_1, c10_1, c2_2, c2_1, c1_2, c1_1


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

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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1


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

(61) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   HELP(x0, s(0), s(s(z0))) -> c1(LT(s(s(z0)), s(0)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)), LT(s(0), s(s(z0))))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2)))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_2, c2_1, c1_2, c1_1, c6_1


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

(65) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, z0, z1, x2) -> c2(PLUS(z0, if(lt(s(x2), z1), z0, z1, s(x2))), HELP(z0, z1, s(x2))) by 
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(s(x2))), s(s(x1))), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) by 
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(s(0)), s(s(x1))), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0)))) by 
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2)))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, IF_4, HELP_3, LT_2

Compound Symbols:   c9_1, c10_1, c2_1, c1_2, c1_1, c6_1, c2_2


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

(71) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, x0, x1, x2) -> c2(HELP(x0, x1, s(x2))) by 
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1


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

(73) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   HELP(x0, s(s(z0)), s(0)) -> c1(IF(true, x0, s(s(z0)), s(0)))
   IF(true, x0, s(s(x1)), s(0)) -> c2(HELP(x0, s(s(x1)), s(s(0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c2(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))), HELP(x0, s(s(x1)), s(s(0))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1


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

(77) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF(true, x0, s(s(x1)), s(0)) -> c(HELP(x0, s(s(x1)), s(s(0))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1


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

(79) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_2, c2_1, c_1


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(PLUS(x0, if(lt(s(s(x2)), s(x1)), x0, s(s(x1)), s(s(s(x2))))), HELP(x0, s(s(x1)), s(s(s(x2))))) by IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1))))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_2, c1_1, c6_1, c2_1, c_1, c2_2


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

(83) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(x0, s(s(z1)), s(s(z0))) -> c1(IF(lt(z0, z1), x0, s(s(z1)), s(s(z0))), LT(s(s(z0)), s(s(z1)))) by 
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
K tuples:
   HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1)))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, HELP_3, LT_2, IF_4

Compound Symbols:   c9_1, c10_1, c1_1, c6_1, c2_1, c_1, c2_2, c1_2


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

(85) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(x0, s(x1), s(x2)) -> c1(LT(s(x2), s(x1))) by 
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c9_1, c10_1, c6_1, c2_1, c_1, c2_2, c1_2, c1_1


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

(87) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0))))) by IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
----------------------------------------

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c9_1, c10_1, c6_1, c2_1, c2_2, c1_2, c1_1, c_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) by 
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2)))))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c9_1, c10_1, c6_1, c2_1, c2_2, c1_2, c1_1, c_1


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

(91) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, x0, s(s(x1)), s(s(x2))) -> c2(HELP(x0, s(s(x1)), s(s(s(x2))))) by 
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c9_1, c10_1, c6_1, c2_2, c1_2, c1_1, c_1, c2_1


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

(93) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, z0, s(s(z1)), s(s(z2))) -> c2(PLUS(z0, if(lt(s(z2), z1), z0, s(s(z1)), s(s(s(z2))))), HELP(z0, s(s(z1)), s(s(s(z2))))) by 
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, HELP_3, IF_4

Compound Symbols:   c9_1, c10_1, c6_1, c1_2, c1_1, c_1, c2_1, c2_2


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

(95) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(IF(lt(s(x2), x1), x0, s(s(x1)), s(s(s(x2)))), LT(s(s(s(x2))), s(s(x1)))) by 
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, HELP_3, IF_4

Compound Symbols:   c9_1, c10_1, c6_1, c1_1, c_1, c2_1, c2_2, c1_2


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace HELP(x0, s(s(x1)), s(s(s(x2)))) -> c1(LT(s(s(s(x2))), s(s(x1)))) by 
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
S tuples:
   PLUS(z0, s(z1)) -> c9(PLUS(z0, z1))
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c9_1, c10_1, c6_1, c_1, c2_1, c2_2, c1_2, c1_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(z1)) -> c9(PLUS(z0, z1)) by 
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
S tuples:
   PLUS(s(z0), z1) -> c10(PLUS(z0, z1))
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_4, HELP_3

Compound Symbols:   c10_1, c6_1, c_1, c2_1, c2_2, c1_2, c1_1, c9_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), z1) -> c10(PLUS(z0, z1)) by 
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
S tuples:
   LT(s(s(y0)), s(s(y1))) -> c6(LT(s(y0), s(y1)))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   LT_2, IF_4, HELP_3, PLUS_2

Compound Symbols:   c6_1, c_1, c2_1, c2_2, c1_2, c1_1, c9_1, c10_1


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

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

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c1_1, c9_1, c10_1, c6_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(LT(s(s(s(s(x2)))), s(s(x1)))) by 
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1)))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c9_1, c10_1, c6_1, c1_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c9(PLUS(z0, s(y1))) by 
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c9_1, c10_1, c6_1, c1_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(y0), s(z1)) -> c9(PLUS(s(y0), z1)) by 
   PLUS(s(z0), s(s(y1))) -> c9(PLUS(s(z0), s(y1)))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(y1)))) -> c9(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(y0)), s(s(y1))) -> c9(PLUS(s(s(y0)), s(y1)))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(y0)), z1) -> c10(PLUS(s(y0), z1)) by 
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(z0), s(s(y1))) -> c10(PLUS(z0, s(s(y1)))) by 
   PLUS(s(s(y0)), s(s(z1))) -> c10(PLUS(s(y0), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(z1))) -> c10(PLUS(s(s(y0)), s(s(z1))))
   PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c10(PLUS(s(y0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(s(y1))))) -> c10(PLUS(s(y0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1)))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c10_1, c6_1, c1_1, c9_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(s(s(y0)), s(y1)) -> c10(PLUS(s(y0), s(y1))) by 
   PLUS(s(s(s(y0))), s(z1)) -> c10(PLUS(s(s(y0)), s(z1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1)))
   PLUS(s(s(s(s(y0)))), s(s(y1))) -> c10(PLUS(s(s(s(y0))), s(s(y1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))
   PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1))))
   PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1))))))
   PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1)))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   help(z0, z1, z2) -> if(lt(z2, z1), z0, z1, z2)
   if(true, z0, z1, z2) -> plus(z0, help(z0, z1, s(z2)))
   if(false, z0, z1, z2) -> 0
   plus(z0, 0) -> z0
   plus(0, z0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, z0, s(s(z1)), s(0)) -> c(PLUS(z0, if(lt(0, z1), z0, s(s(z1)), s(s(0)))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1)))
   PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1))))
   PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1))))))
   PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1)))))
S tuples:
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   IF(true, x0, s(s(x1)), s(s(s(x2)))) -> c2(PLUS(x0, if(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2)))))), HELP(x0, s(s(x1)), s(s(s(s(x2))))))
   HELP(x0, s(s(x1)), s(s(s(s(x2))))) -> c1(IF(lt(s(s(x2)), x1), x0, s(s(x1)), s(s(s(s(x2))))), LT(s(s(s(s(x2)))), s(s(x1))))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c6(LT(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c9(PLUS(z0, s(s(y1))))
   PLUS(s(y0), s(s(z1))) -> c9(PLUS(s(y0), s(z1)))
   PLUS(s(s(y0)), s(s(z1))) -> c9(PLUS(s(s(y0)), s(z1)))
   PLUS(s(y0), s(s(s(y1)))) -> c9(PLUS(s(y0), s(s(y1))))
   PLUS(s(s(y0)), s(z1)) -> c9(PLUS(s(s(y0)), z1))
   PLUS(s(z0), s(s(s(s(y1))))) -> c9(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(y0)), s(s(s(y1)))) -> c9(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(s(y0))), z1) -> c10(PLUS(s(s(y0)), z1))
   PLUS(s(s(z0)), s(s(y1))) -> c10(PLUS(s(z0), s(s(y1))))
   PLUS(s(s(s(y0))), s(y1)) -> c10(PLUS(s(s(y0)), s(y1)))
   PLUS(s(s(z0)), s(s(s(y1)))) -> c10(PLUS(s(z0), s(s(s(y1)))))
   PLUS(s(s(s(y0))), s(s(y1))) -> c10(PLUS(s(s(y0)), s(s(y1))))
   PLUS(s(s(z0)), s(s(s(s(y1))))) -> c10(PLUS(s(z0), s(s(s(s(y1))))))
   PLUS(s(s(s(y0))), s(s(s(y1)))) -> c10(PLUS(s(s(y0)), s(s(s(y1)))))
   PLUS(s(z0), s(s(s(y1)))) -> c10(PLUS(z0, s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(s(z1))) -> c10(PLUS(s(s(s(y0))), s(s(z1))))
   PLUS(s(s(s(y0))), s(s(s(s(y1))))) -> c10(PLUS(s(s(y0)), s(s(s(s(y1))))))
   PLUS(s(s(s(s(y0)))), s(s(s(y1)))) -> c10(PLUS(s(s(s(y0))), s(s(s(y1)))))
   PLUS(s(s(s(s(y0)))), s(z1)) -> c10(PLUS(s(s(s(y0))), s(z1)))
   PLUS(s(s(s(s(s(y0))))), s(s(y1))) -> c10(PLUS(s(s(s(s(y0)))), s(s(y1))))
   PLUS(s(s(s(s(y0)))), s(s(s(s(y1))))) -> c10(PLUS(s(s(s(y0))), s(s(s(s(y1))))))
   PLUS(s(s(s(s(s(y0))))), s(s(s(y1)))) -> c10(PLUS(s(s(s(s(y0)))), s(s(s(y1)))))
K tuples:
   IF(true, x0, s(s(x1)), s(0)) -> c(PLUS(x0, if(lt(s(0), s(x1)), x0, s(s(x1)), s(s(0)))))
   HELP(z0, s(s(s(y1))), s(s(s(s(z2))))) -> c1(LT(s(s(s(s(z2)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, help_3, if_4, plus_2

Defined Pair Symbols:   IF_4, HELP_3, LT_2, PLUS_2

Compound Symbols:   c_1, c2_1, c2_2, c1_2, c6_1, c1_1, c9_1, c10_1