KILLED
proof of input_dmPkop8uwa.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 3 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 326 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 119 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 2843 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 529 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 [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (48) CdtProblem
    (49) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (66) CdtProblem
    (67) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 24 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (82) CdtProblem
    (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) 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) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(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) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0')
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0'
   sum(cons(0', xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(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) -> sum(generate(x, y))
   generate(x, y) -> gen(x, y, 0)
   gen(x, y, z) -> if(ge(z, x), x, y, z)
   if(true, x, y, z) -> nil
   if(false, x, y, z) -> cons(y, gen(x, y, s(z)))
   sum(nil) -> 0
   sum(cons(0, xs)) -> sum(xs)
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(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) -> sum(generate(x, y)) [1]
   generate(x, y) -> gen(x, y, 0) [1]
   gen(x, y, z) -> if(ge(z, x), x, y, z) [1]
   if(true, x, y, z) -> nil [1]
   if(false, x, y, z) -> cons(y, gen(x, y, s(z))) [1]
   sum(nil) -> 0 [1]
   sum(cons(0, xs)) -> sum(xs) [1]
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(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) -> sum(generate(x, y)) [1]
   generate(x, y) -> gen(x, y, 0) [1]
   gen(x, y, z) -> if(ge(z, x), x, y, z) [1]
   if(true, x, y, z) -> nil [1]
   if(false, x, y, z) -> cons(y, gen(x, y, s(z))) [1]
   sum(nil) -> 0 [1]
   sum(cons(0, xs)) -> sum(xs) [1]
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
sum :: nil:cons -> 0:s
generate :: 0:s -> 0:s -> nil:cons
gen :: 0:s -> 0:s -> 0:s -> nil:cons
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> nil:cons
ge :: 0:s -> 0:s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0:s -> nil:cons -> nil:cons
s :: 0:s -> 0:s

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

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

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

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

   times_2
   sum_1

(c) The following functions are completely defined:

   ge_2
   generate_2
   gen_3
   if_4

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   times(x, y) -> sum(generate(x, y)) [1]
   generate(x, y) -> gen(x, y, 0) [1]
   gen(x, y, z) -> if(ge(z, x), x, y, z) [1]
   if(true, x, y, z) -> nil [1]
   if(false, x, y, z) -> cons(y, gen(x, y, s(z))) [1]
   sum(nil) -> 0 [1]
   sum(cons(0, xs)) -> sum(xs) [1]
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
sum :: nil:cons -> 0:s
generate :: 0:s -> 0:s -> nil:cons
gen :: 0:s -> 0:s -> 0:s -> nil:cons
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> nil:cons
ge :: 0:s -> 0:s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0:s -> nil:cons -> nil:cons
s :: 0:s -> 0:s

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

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

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

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

   times(x, y) -> sum(gen(x, y, 0)) [2]
   generate(x, y) -> gen(x, y, 0) [1]
   gen(0, y, z) -> if(true, 0, y, z) [2]
   gen(s(y'), y, 0) -> if(false, s(y'), y, 0) [2]
   gen(s(y''), y, s(x')) -> if(ge(x', y''), s(y''), y, s(x')) [2]
   if(true, x, y, z) -> nil [1]
   if(false, x, y, z) -> cons(y, gen(x, y, s(z))) [1]
   sum(nil) -> 0 [1]
   sum(cons(0, xs)) -> sum(xs) [1]
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
sum :: nil:cons -> 0:s
generate :: 0:s -> 0:s -> nil:cons
gen :: 0:s -> 0:s -> 0:s -> nil:cons
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> nil:cons
ge :: 0:s -> 0:s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0:s -> nil:cons -> nil:cons
s :: 0:s -> 0:s

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

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

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

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

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

   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(x', y''), 1 + y'', y, 1 + x') :|: z' = 1 + y'', z'' = y, y >= 0, x' >= 0, y'' >= 0, z1 = 1 + x'
   gen(z', z'', z1) -{ 2 }-> if(1, 0, y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + y', y, 0) :|: z1 = 0, z'' = y, y >= 0, y' >= 0, z' = 1 + y'
   generate(z', z'') -{ 1 }-> gen(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + y + gen(x, y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   sum(z') -{ 1 }-> sum(xs) :|: xs >= 0, z' = 1 + 0 + xs
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(x, y, 0)) :|: z' = x, z'' = y, x >= 0, y >= 0


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

(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:

   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(z1 - 1, z' - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0


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

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

   { ge }
   { sum }
   { if, gen }
   { generate }
   { times }

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

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

   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(z1 - 1, z' - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {sum}, {if,gen}, {generate}, {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:

   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(z1 - 1, z' - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {sum}, {if,gen}, {generate}, {times}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(z1 - 1, z' - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {sum}, {if,gen}, {generate}, {times}
Previous analysis results are:
  ge: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(ge(z1 - 1, z' - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

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

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

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 }-> sum(z' - 1) :|: z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 3 + x + xs }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + x + xs, xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

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

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

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 3 + x + xs }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + x + xs, xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   gen(z', z'', z1) -{ 3 + z' }-> if(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(1, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(0, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   sum(z') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1, z' - 1 >= 0
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 3 + x + xs }-> 1 + s1 :|: s1 >= 0, s1 <= 1 + x + xs, xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

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

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

(37) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:
none

And the following fresh constants: none

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

(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) -> sum(generate(x, y)) [1]
   generate(x, y) -> gen(x, y, 0) [1]
   gen(x, y, z) -> if(ge(z, x), x, y, z) [1]
   if(true, x, y, z) -> nil [1]
   if(false, x, y, z) -> cons(y, gen(x, y, s(z))) [1]
   sum(nil) -> 0 [1]
   sum(cons(0, xs)) -> sum(xs) [1]
   sum(cons(s(x), xs)) -> s(sum(cons(x, xs))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]

The TRS has the following type information:
times :: 0:s -> 0:s -> 0:s
sum :: nil:cons -> 0:s
generate :: 0:s -> 0:s -> nil:cons
gen :: 0:s -> 0:s -> 0:s -> nil:cons
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> nil:cons
ge :: 0:s -> 0:s -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0:s -> nil:cons -> nil:cons
s :: 0:s -> 0:s

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 => 1
   nil => 0
   false => 0

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

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

   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 0 :|: y >= 0, z'' = 1 + y, z' = 0
   gen(z', z'', z1) -{ 1 }-> if(ge(z, x), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   generate(z', z'') -{ 1 }-> gen(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 1 }-> 1 + y + gen(x, y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   sum(z') -{ 1 }-> sum(xs) :|: xs >= 0, z' = 1 + 0 + xs
   sum(z') -{ 1 }-> 0 :|: z' = 0
   sum(z') -{ 1 }-> 1 + sum(1 + x + xs) :|: xs >= 0, x >= 0, z' = 1 + (1 + x) + xs
   times(z', z'') -{ 1 }-> sum(generate(x, y)) :|: z' = x, z'' = y, x >= 0, y >= 0

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) -> sum(generate(z0, z1))
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(nil) -> c5
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(z0, 0) -> c8
   GE(0, s(z0)) -> c9
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
S tuples:
   TIMES(z0, z1) -> c(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(nil) -> c5
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(z0, 0) -> c8
   GE(0, s(z0)) -> c9
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, ge_2

Defined Pair Symbols:   TIMES_2, GENERATE_2, GEN_3, IF_4, SUM_1, GE_2

Compound Symbols:   c_2, c1_1, c2_2, c3, c4_1, c5, c6_1, c7_1, c8, c9, c10_1


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   GE(z0, 0) -> c8
   IF(true, z0, z1, z2) -> c3
   GE(0, s(z0)) -> c9
   SUM(nil) -> c5

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> sum(generate(z0, z1))
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   TIMES(z0, z1) -> c(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
S tuples:
   TIMES(z0, z1) -> c(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, ge_2

Defined Pair Symbols:   TIMES_2, GENERATE_2, GEN_3, IF_4, SUM_1, GE_2

Compound Symbols:   c_2, c1_1, c2_2, c4_1, c6_1, c7_1, c10_1


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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> sum(generate(z0, z1))
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   TIMES(z0, z1) -> c3(GENERATE(z0, z1))
S tuples:
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   TIMES(z0, z1) -> c3(GENERATE(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, ge_2

Defined Pair Symbols:   GENERATE_2, GEN_3, IF_4, SUM_1, GE_2, TIMES_2

Compound Symbols:   c1_1, c2_2, c4_1, c6_1, c7_1, c10_1, c3_1


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

(47) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   TIMES(z0, z1) -> c3(GENERATE(z0, z1))
   GENERATE(z0, z1) -> c1(GEN(z0, z1, 0))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> sum(generate(z0, z1))
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, ge_2

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, GE_2, TIMES_2

Compound Symbols:   c2_2, c4_1, c6_1, c7_1, c10_1, c3_1


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

(49) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   times(z0, z1) -> sum(generate(z0, z1))
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
Tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, ge_2

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, GE_2, TIMES_2

Compound Symbols:   c2_2, c4_1, c6_1, c7_1, c10_1, c3_1


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

(51) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   times(z0, z1) -> sum(generate(z0, z1))
   sum(nil) -> 0
   sum(cons(0, z0)) -> sum(z0)
   sum(cons(s(z0), z1)) -> s(sum(cons(z0, z1)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, generate_2, gen_3, if_4

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, GE_2, TIMES_2

Compound Symbols:   c2_2, c4_1, c6_1, c7_1, c10_1, c3_1


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

(53) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(z0, z1, z2) -> c2(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0)) by 
   GEN(0, x1, z0) -> c2(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c2(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(0, x1, z0) -> c2(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c2(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(0, x1, z0) -> c2(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c2(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2


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

(55) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GEN(s(z0), x1, 0) -> c2(IF(false, s(z0), x1, 0), GE(0, s(z0)))
Removed 1 trailing nodes:
   GEN(0, x1, z0) -> c2(IF(true, 0, x1, z0), GE(z0, 0))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c3(SUM(generate(z0, z1))) by 
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   generate(z0, z1) -> gen(z0, z1, 0)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c3_1


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

(59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   generate(z0, z1) -> gen(z0, z1, 0)

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c3_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(s(z1), x1, s(z0)) -> c2(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1))) by 
   GEN(s(0), x1, s(z0)) -> c2(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(0), x1, s(z0)) -> c2(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(0), x1, s(z0)) -> c2(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2, c2_1


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2, c2_1


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

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(GE(x_1, x_2)) = 0
   POL(GEN(x_1, x_2, x_3)) = x_1
   POL(IF(x_1, x_2, x_3, x_4)) = x_2
   POL(SUM(x_1)) = [1]
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1] + x_2
   POL(gen(x_1, x_2, x_3)) = x_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(nil) = [1]
   POL(s(x_1)) = x_1
   POL(true) = [1]

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2, c2_1


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

(67) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
We considered the (Usable) Rules:none
And the Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(GE(x_1, x_2)) = 0
   POL(GEN(x_1, x_2, x_3)) = [1] + x_1
   POL(IF(x_1, x_2, x_3, x_4)) = [1] + x_2
   POL(SUM(x_1)) = [1]
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_1
   POL(false) = [1]
   POL(ge(x_1, x_2)) = x_2
   POL(gen(x_1, x_2, x_3)) = [1] + x_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(nil) = [1]
   POL(s(x_1)) = x_1
   POL(true) = [1]

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, TIMES_2, GEN_3

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c3_1, c2_2, c2_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c3(SUM(gen(z0, z1, 0))) by 
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c2_1, c3_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(s(s(z1)), x1, s(s(z0))) -> c2(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1)))) by 
   GEN(s(s(0)), x1, s(s(z0))) -> c2(IF(true, s(s(0)), x1, s(s(z0))), GE(s(s(z0)), s(s(0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(IF(true, s(s(0)), x1, s(s(z0))), GE(s(s(z0)), s(s(0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(IF(true, s(s(0)), x1, s(s(z0))), GE(s(s(z0)), s(s(0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c2_1, c3_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c2_1, c3_1


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   ge(z0, 0) -> true
And the Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(GE(x_1, x_2)) = 0
   POL(GEN(x_1, x_2, x_3)) = [1] + x_1 + x_3
   POL(IF(x_1, x_2, x_3, x_4)) = x_1 + x_2
   POL(SUM(x_1)) = [1]
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(if(x_1, x_2, x_3, x_4)) = x_2 + x_3 + x_4
   POL(nil) = 0
   POL(s(x_1)) = 0
   POL(true) = [1]

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c2_1, c3_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c3(SUM(if(ge(0, z0), z0, z1, 0))) by 
   TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
   TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2)))
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, GE_2, GEN_3, TIMES_2

Compound Symbols:   c4_1, c6_1, c7_1, c10_1, c2_2, c2_1, c3_1


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

(79) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, z0, z1, z2) -> c4(GEN(z0, z1, s(z2))) by 
   IF(false, s(s(x0)), x1, s(0)) -> c4(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
   TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(x0)), x1, s(0)) -> c4(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   IF(false, s(s(x0)), x1, s(0)) -> c4(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
K tuples:
   TIMES(z0, z1) -> c3(SUM(generate(z0, z1)))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_2, c2_1, c3_1, c4_1


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

(81) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 6 leading nodes:
   GEN(s(s(z0)), x1, s(0)) -> c2(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   IF(false, s(s(x0)), x1, s(0)) -> c4(GEN(s(s(x0)), x1, s(s(0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c2(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   GEN(s(0), x1, s(z0)) -> c2(GE(s(z0), s(0)))
   GEN(s(s(0)), x1, s(s(z0))) -> c2(GE(s(s(z0)), s(s(0))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
K tuples:
   GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_1, c2_2, c3_1, c4_1


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

(83) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(x0), x1, s(x2)) -> c2(GE(s(x2), s(x0))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
K tuples:
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_2, c2_1, c3_1, c4_1


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace TIMES(0, x1) -> c3(SUM(if(true, 0, x1, 0))) by TIMES(0, z0) -> c3(SUM(nil))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(0, z0) -> c3(SUM(nil))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
K tuples:
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_2, c2_1, c3_1, c4_1


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

(87) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   TIMES(0, z0) -> c3(SUM(nil))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
K tuples:
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_2, c2_1, c3_1, c4_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c2(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1))))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0)))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
K tuples:
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_1, c3_1, c4_1, c2_2


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

(91) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace TIMES(s(z0), x1) -> c3(SUM(if(false, s(z0), x1, 0))) by TIMES(s(z0), z1) -> c3(SUM(cons(z1, gen(s(z0), z1, s(0)))))
----------------------------------------

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, gen(s(z0), z1, s(0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
K tuples:
   GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, IF_4, TIMES_2

Compound Symbols:   c6_1, c7_1, c10_1, c2_1, c4_1, c2_2, c3_1


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

(93) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(s(x0)), x1, s(s(x2))) -> c2(GE(s(s(x2)), s(s(x0)))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, gen(s(z0), z1, s(0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, IF_4, GEN_3, TIMES_2

Compound Symbols:   c6_1, c7_1, c10_1, c4_1, c2_1, c2_2, c3_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace TIMES(s(z0), z1) -> c3(SUM(cons(z1, gen(s(z0), z1, s(0))))) by TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, IF_4, GEN_3, TIMES_2

Compound Symbols:   c6_1, c7_1, c10_1, c4_1, c2_1, c2_2, c3_1


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

(97) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, s(s(s(x0))), x1, s(s(s(x2)))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(x2)))))) by 
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
S tuples:
   SUM(cons(0, z0)) -> c6(SUM(z0))
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c6_1, c7_1, c10_1, c2_1, c2_2, c3_1, c4_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(0, z0)) -> c6(SUM(z0)) by 
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
S tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c7_1, c10_1, c2_1, c2_2, c3_1, c4_1, c6_1


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

(101) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0))))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
S tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c7_1, c10_1, c2_1, c2_2, c3_1, c4_1, c6_1


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

(103) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(IF(ge(s(x2), x0), s(s(s(x0))), x1, s(s(s(s(x2))))), GE(s(s(s(s(x2)))), s(s(s(x0))))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
S tuples:
   SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1)))
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   SUM_1, GE_2, GEN_3, TIMES_2, IF_4

Compound Symbols:   c7_1, c10_1, c2_1, c3_1, c4_1, c6_1, c2_2


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(s(z0), z1)) -> c7(SUM(cons(z0, z1))) by 
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0)))))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
S tuples:
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   GE_2, GEN_3, TIMES_2, IF_4, SUM_1

Compound Symbols:   c10_1, c2_1, c3_1, c4_1, c6_1, c2_2, c7_1


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

(107) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace GEN(s(s(s(x0))), x1, s(s(s(s(x2))))) -> c2(GE(s(s(s(s(x2)))), s(s(s(x0))))) by 
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
S tuples:
   GE(s(z0), s(z1)) -> c10(GE(z0, z1))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   GE_2, TIMES_2, IF_4, SUM_1, GEN_3

Compound Symbols:   c10_1, c3_1, c4_1, c6_1, c2_1, c2_2, c7_1


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

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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0)))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, SUM_1, GEN_3, GE_2

Compound Symbols:   c3_1, c4_1, c6_1, c2_1, c2_2, c7_1, c10_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(0, cons(0, y0))) -> c6(SUM(cons(0, y0))) by 
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1)))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, SUM_1, GEN_3, GE_2

Compound Symbols:   c3_1, c4_1, c6_1, c2_1, c2_2, c7_1, c10_1


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

(113) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(0, cons(s(y0), y1))) -> c6(SUM(cons(s(y0), y1))) by 
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1)))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, GEN_3, SUM_1, GE_2

Compound Symbols:   c3_1, c4_1, c2_1, c2_2, c7_1, c10_1, c6_1


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

(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(s(s(y0)), z1)) -> c7(SUM(cons(s(y0), z1))) by 
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, GEN_3, SUM_1, GE_2

Compound Symbols:   c3_1, c4_1, c2_1, c2_2, c7_1, c10_1, c6_1


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

(117) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(s(0), cons(0, y0))) -> c7(SUM(cons(0, cons(0, y0)))) by 
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, GEN_3, SUM_1, GE_2

Compound Symbols:   c3_1, c4_1, c2_1, c2_2, c7_1, c10_1, c6_1


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

(119) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(cons(s(0), cons(s(y0), y1))) -> c7(SUM(cons(0, cons(s(y0), y1)))) by 
   SUM(cons(s(0), cons(s(s(y0)), z1))) -> c7(SUM(cons(0, cons(s(s(y0)), z1))))
   SUM(cons(s(0), cons(s(0), cons(0, y0)))) -> c7(SUM(cons(0, cons(s(0), cons(0, y0)))))
   SUM(cons(s(0), cons(s(0), cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(s(0), cons(s(y0), y1)))))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
   SUM(cons(s(0), cons(s(s(y0)), z1))) -> c7(SUM(cons(0, cons(s(s(y0)), z1))))
   SUM(cons(s(0), cons(s(0), cons(0, y0)))) -> c7(SUM(cons(0, cons(s(0), cons(0, y0)))))
   SUM(cons(s(0), cons(s(0), cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(s(0), cons(s(y0), y1)))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1)))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
   SUM(cons(s(0), cons(s(s(y0)), z1))) -> c7(SUM(cons(0, cons(s(s(y0)), z1))))
   SUM(cons(s(0), cons(s(0), cons(0, y0)))) -> c7(SUM(cons(0, cons(s(0), cons(0, y0)))))
   SUM(cons(s(0), cons(s(0), cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(s(0), cons(s(y0), y1)))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, GEN_3, GE_2, SUM_1

Compound Symbols:   c3_1, c4_1, c2_1, c2_2, c10_1, c6_1, c7_1


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

(121) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c10(GE(s(y0), s(y1))) by 
   GE(s(s(s(y0))), s(s(s(y1)))) -> c10(GE(s(s(y0)), s(s(y1))))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(true, z0, z1, z2) -> nil
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
Tuples:
   TIMES(s(z0), z1) -> c3(SUM(cons(z1, if(ge(s(0), s(z0)), s(z0), z1, s(0)))))
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
   SUM(cons(s(0), cons(s(s(y0)), z1))) -> c7(SUM(cons(0, cons(s(s(y0)), z1))))
   SUM(cons(s(0), cons(s(0), cons(0, y0)))) -> c7(SUM(cons(0, cons(s(0), cons(0, y0)))))
   SUM(cons(s(0), cons(s(0), cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(s(0), cons(s(y0), y1)))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c10(GE(s(s(y0)), s(s(y1))))
S tuples:
   IF(false, s(s(s(x0))), x1, s(s(s(s(x2))))) -> c4(GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))))
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(IF(ge(s(s(x2)), x0), s(s(s(x0))), x1, s(s(s(s(s(x2)))))), GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
   SUM(cons(0, cons(0, cons(0, y0)))) -> c6(SUM(cons(0, cons(0, y0))))
   SUM(cons(0, cons(0, cons(s(y0), y1)))) -> c6(SUM(cons(0, cons(s(y0), y1))))
   SUM(cons(0, cons(s(s(y0)), z1))) -> c6(SUM(cons(s(s(y0)), z1)))
   SUM(cons(0, cons(s(0), cons(0, y0)))) -> c6(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(0, cons(s(0), cons(s(y0), y1)))) -> c6(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(s(s(y0))), z1)) -> c7(SUM(cons(s(s(y0)), z1)))
   SUM(cons(s(s(0)), cons(0, y0))) -> c7(SUM(cons(s(0), cons(0, y0))))
   SUM(cons(s(s(0)), cons(s(y0), y1))) -> c7(SUM(cons(s(0), cons(s(y0), y1))))
   SUM(cons(s(0), cons(0, cons(0, y0)))) -> c7(SUM(cons(0, cons(0, cons(0, y0)))))
   SUM(cons(s(0), cons(0, cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(0, cons(s(y0), y1)))))
   SUM(cons(s(0), cons(s(s(y0)), z1))) -> c7(SUM(cons(0, cons(s(s(y0)), z1))))
   SUM(cons(s(0), cons(s(0), cons(0, y0)))) -> c7(SUM(cons(0, cons(s(0), cons(0, y0)))))
   SUM(cons(s(0), cons(s(0), cons(s(y0), y1)))) -> c7(SUM(cons(0, cons(s(0), cons(s(y0), y1)))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c10(GE(s(s(y0)), s(s(y1))))
K tuples:
   GEN(s(s(s(x0))), x1, s(s(s(s(s(x2)))))) -> c2(GE(s(s(s(s(s(x2))))), s(s(s(x0)))))
Defined Rule Symbols:   ge_2, gen_3, if_4

Defined Pair Symbols:   TIMES_2, IF_4, GEN_3, SUM_1, GE_2

Compound Symbols:   c3_1, c4_1, c2_1, c2_2, c6_1, c7_1, c10_1