KILLED
proof of input_YV64cxFKOt.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 11 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 252 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 80 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 428 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 160 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 71 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 201 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 85 ms]
        (54) CpxRNTS
        (55) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (56) CpxRNTS
        (57) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
        (58) CpxRNTS
        (59) IntTrsBoundProof [UPPER BOUND(ID), 11 ms]
        (60) CpxRNTS
        (61) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (62) CpxRNTS
        (63) IntTrsBoundProof [UPPER BOUND(ID), 17.5 s]
        (64) CpxRNTS
        (65) IntTrsBoundProof [UPPER BOUND(ID), 13.9 s]
        (66) CpxRNTS
    (67) CompletionProof [UPPER BOUND(ID), 0 ms]
    (68) CpxTypedWeightedCompleteTrs
        (69) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 14 ms]
        (70) CpxRNTS
(71) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (80) CdtProblem
    (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 4 ms]
    (90) CdtProblem
    (91) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 51 ms]
    (98) CdtProblem
    (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 28 ms]
    (100) CdtProblem
    (101) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 42 ms]
    (116) CdtProblem
    (117) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 47 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (122) CdtProblem
    (123) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms]
    (136) CdtProblem
    (137) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 57 ms]
    (154) CdtProblem
    (155) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms]
    (156) CdtProblem
    (157) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 43 ms]
    (188) CdtProblem
    (189) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (202) CdtProblem
    (203) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (210) CdtProblem
    (211) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (214) CdtProblem
    (215) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (216) CdtProblem
    (217) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (218) CdtProblem
    (219) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) 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(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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(xs) -> sum2(xs, 0')
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0') -> true
   isZero(s(0')) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0') -> s(s(0'))
   p(s(0')) -> 0'
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0') -> true
   ge(0', s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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(xs) -> sum2(xs, 0)
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y)
   ifsum(true, b, xs, y) -> y
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y)
   ifsum2(true, xs, y) -> sum2(tail(xs), y)
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y))
   isNil(nil) -> true
   isNil(cons(x, xs)) -> false
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   head(cons(x, xs)) -> x
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(x))) -> isZero(s(x))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(x))) -> s(p(s(x)))
   ge(x, 0) -> true
   ge(0, s(y)) -> false
   ge(s(x), s(y)) -> ge(x, y)
   a -> c
   a -> d

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(xs) -> sum2(xs, 0) [1]
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) [1]
   ifsum(true, b, xs, y) -> y [1]
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y) [1]
   ifsum2(true, xs, y) -> sum2(tail(xs), y) [1]
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) [1]
   isNil(nil) -> true [1]
   isNil(cons(x, xs)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   head(cons(x, xs)) -> x [1]
   head(nil) -> error [1]
   isZero(0) -> true [1]
   isZero(s(0)) -> false [1]
   isZero(s(s(x))) -> isZero(s(x)) [1]
   p(0) -> s(s(0)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> c [1]
   a -> d [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(xs) -> sum2(xs, 0) [1]
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) [1]
   ifsum(true, b, xs, y) -> y [1]
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y) [1]
   ifsum2(true, xs, y) -> sum2(tail(xs), y) [1]
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) [1]
   isNil(nil) -> true [1]
   isNil(cons(x, xs)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   head(cons(x, xs)) -> x [1]
   head(nil) -> error [1]
   isZero(0) -> true [1]
   isZero(s(0)) -> false [1]
   isZero(s(s(x))) -> isZero(s(x)) [1]
   p(0) -> s(s(0)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> c [1]
   a -> d [1]

The TRS has the following type information:
times :: 0:s:error -> 0:s:error -> 0:s:error
sum :: nil:cons -> 0:s:error
generate :: 0:s:error -> 0:s:error -> nil:cons
gen :: 0:s:error -> 0:s:error -> 0:s:error -> nil:cons
0 :: 0:s:error
if :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> nil:cons
ge :: 0:s:error -> 0:s:error -> true:false
true :: true:false
nil :: nil:cons
false :: true:false
cons :: 0:s:error -> nil:cons -> nil:cons
s :: 0:s:error -> 0:s:error
sum2 :: nil:cons -> 0:s:error -> 0:s:error
ifsum :: true:false -> true:false -> nil:cons -> 0:s:error -> 0:s:error
isNil :: nil:cons -> true:false
isZero :: 0:s:error -> true:false
head :: nil:cons -> 0:s:error
ifsum2 :: true:false -> nil:cons -> 0:s:error -> 0:s:error
tail :: nil:cons -> nil:cons
p :: 0:s:error -> 0:s:error
error :: 0:s:error
a :: c:d
c :: c:d
d :: c:d

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
   sum2_2
   ifsum_4
   ifsum2_3
   a

(c) The following functions are completely defined:

   tail_1
   p_1
   head_1
   isNil_1
   isZero_1
   ge_2
   generate_2
   gen_3
   if_4

Due to the following rules being added:

   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3) -> nil [0]

And the following fresh constants:    null_p, null_isZero, null_ge

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

(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(xs) -> sum2(xs, 0) [1]
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) [1]
   ifsum(true, b, xs, y) -> y [1]
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y) [1]
   ifsum2(true, xs, y) -> sum2(tail(xs), y) [1]
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) [1]
   isNil(nil) -> true [1]
   isNil(cons(x, xs)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   head(cons(x, xs)) -> x [1]
   head(nil) -> error [1]
   isZero(0) -> true [1]
   isZero(s(0)) -> false [1]
   isZero(s(s(x))) -> isZero(s(x)) [1]
   p(0) -> s(s(0)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> c [1]
   a -> d [1]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
times :: 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p
sum :: nil:cons -> 0:s:error:null_p
generate :: 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
gen :: 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
0 :: 0:s:error:null_p
if :: true:false:null_isZero:null_ge -> 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
ge :: 0:s:error:null_p -> 0:s:error:null_p -> true:false:null_isZero:null_ge
true :: true:false:null_isZero:null_ge
nil :: nil:cons
false :: true:false:null_isZero:null_ge
cons :: 0:s:error:null_p -> nil:cons -> nil:cons
s :: 0:s:error:null_p -> 0:s:error:null_p
sum2 :: nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
ifsum :: true:false:null_isZero:null_ge -> true:false:null_isZero:null_ge -> nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
isNil :: nil:cons -> true:false:null_isZero:null_ge
isZero :: 0:s:error:null_p -> true:false:null_isZero:null_ge
head :: nil:cons -> 0:s:error:null_p
ifsum2 :: true:false:null_isZero:null_ge -> nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
tail :: nil:cons -> nil:cons
p :: 0:s:error:null_p -> 0:s:error:null_p
error :: 0:s:error:null_p
a :: c:d
c :: c:d
d :: c:d
null_p :: 0:s:error:null_p
null_isZero :: true:false:null_isZero:null_ge
null_ge :: true:false:null_isZero:null_ge

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]
   gen(x, y, z) -> if(null_ge, 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(xs) -> sum2(xs, 0) [1]
   sum2(nil, y) -> ifsum(true, isZero(error), nil, y) [3]
   sum2(cons(x'', xs'), y) -> ifsum(false, isZero(x''), cons(x'', xs'), y) [3]
   ifsum(true, b, xs, y) -> y [1]
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y) [1]
   ifsum2(true, nil, y) -> sum2(nil, y) [2]
   ifsum2(true, cons(x1, xs''), y) -> sum2(xs'', y) [2]
   ifsum2(false, cons(x2, xs1), y) -> sum2(cons(p(x2), xs1), s(y)) [3]
   ifsum2(false, nil, y) -> sum2(cons(p(error), nil), s(y)) [3]
   isNil(nil) -> true [1]
   isNil(cons(x, xs)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   head(cons(x, xs)) -> x [1]
   head(nil) -> error [1]
   isZero(0) -> true [1]
   isZero(s(0)) -> false [1]
   isZero(s(s(x))) -> isZero(s(x)) [1]
   p(0) -> s(s(0)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> c [1]
   a -> d [1]
   p(v0) -> null_p [0]
   isZero(v0) -> null_isZero [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3) -> nil [0]

The TRS has the following type information:
times :: 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p
sum :: nil:cons -> 0:s:error:null_p
generate :: 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
gen :: 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
0 :: 0:s:error:null_p
if :: true:false:null_isZero:null_ge -> 0:s:error:null_p -> 0:s:error:null_p -> 0:s:error:null_p -> nil:cons
ge :: 0:s:error:null_p -> 0:s:error:null_p -> true:false:null_isZero:null_ge
true :: true:false:null_isZero:null_ge
nil :: nil:cons
false :: true:false:null_isZero:null_ge
cons :: 0:s:error:null_p -> nil:cons -> nil:cons
s :: 0:s:error:null_p -> 0:s:error:null_p
sum2 :: nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
ifsum :: true:false:null_isZero:null_ge -> true:false:null_isZero:null_ge -> nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
isNil :: nil:cons -> true:false:null_isZero:null_ge
isZero :: 0:s:error:null_p -> true:false:null_isZero:null_ge
head :: nil:cons -> 0:s:error:null_p
ifsum2 :: true:false:null_isZero:null_ge -> nil:cons -> 0:s:error:null_p -> 0:s:error:null_p
tail :: nil:cons -> nil:cons
p :: 0:s:error:null_p -> 0:s:error:null_p
error :: 0:s:error:null_p
a :: c:d
c :: c:d
d :: c:d
null_p :: 0:s:error:null_p
null_isZero :: true:false:null_isZero:null_ge
null_ge :: true:false:null_isZero:null_ge

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

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

   0 => 0
   true => 2
   nil => 0
   false => 1
   error => 1
   c => 0
   d => 1
   null_p => 0
   null_isZero => 0
   null_ge => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   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(2, 0, y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + y', y, 0) :|: z1 = 0, z'' = y, y >= 0, y' >= 0, z' = 1 + y'
   gen(z', z'', z1) -{ 1 }-> if(0, 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
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   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' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> y :|: b >= 0, xs >= 0, z2 = y, z'' = b, z' = 2, y >= 0, z1 = xs
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(b, xs, y) :|: b >= 0, xs >= 0, z2 = y, z'' = b, y >= 0, z' = 1, z1 = xs
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', y) :|: z1 = y, x1 >= 0, z' = 2, xs'' >= 0, y >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, y) :|: z'' = 0, z1 = y, z' = 2, y >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + y) :|: z1 = y, z'' = 1 + x2 + xs1, y >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + y) :|: z'' = 0, z1 = y, y >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(xs, 0) :|: xs >= 0, z' = xs
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, y) :|: z'' = y, y >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', y) :|: z'' = y, xs' >= 0, y >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + (z' - 2)) :|: z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, z'') :|: z'' >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', z'') :|: xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   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:

   { isZero }
   { tail }
   { ge }
   { head }
   { isNil }
   { p }
   { a }
   { ifsum2, ifsum, sum2 }
   { if, gen }
   { sum }
   { generate }
   { times }

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + (z' - 2)) :|: z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, z'') :|: z'' >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', z'') :|: xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isZero}, {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + (z' - 2)) :|: z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, z'') :|: z'' >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', z'') :|: xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isZero}, {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + (z' - 2)) :|: z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, z'') :|: z'' >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', z'') :|: xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isZero}, {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: isZero
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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 1 }-> isZero(1 + (z' - 2)) :|: z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 3 }-> ifsum(2, isZero(1), 0, z'') :|: z'' >= 0, z' = 0
   sum2(z', z'') -{ 3 }-> ifsum(1, isZero(x''), 1 + x'' + xs', z'') :|: xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: tail
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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {tail}, {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: ?, size: O(n^1) [z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], 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:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']

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

(33) 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: 2

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ge}, {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: ?, size: O(1) [2]

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

(35) 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'

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 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(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {head}, {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: ?, size: O(n^1) [1 + z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {isNil}, {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: ?, size: O(1) [2]

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]

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

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {p}, {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: ?, size: O(n^1) [2 + z']

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(54)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(x2) + xs1, 1 + z1) :|: z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 3 }-> sum2(1 + p(1) + 0, 1 + z1) :|: z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 1 + p(1 + (z' - 2)) :|: z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']

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

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

(56)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(58)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {a}, {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']
  a: runtime: ?, size: O(1) [1]

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

(59) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(60)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

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

(62)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: ifsum2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

Computed SIZE bound using CoFloCo for: sum2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(64)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  ifsum2: runtime: ?, size: INF
  ifsum: runtime: ?, size: INF
  sum2: runtime: ?, size: INF

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

(65) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: ifsum2
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

(66)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 2 + z' }-> s2 :|: s2 >= 0, s2 <= 2, z' - 1 >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   gen(z', z'', z1) -{ 3 + z1 }-> if(s1, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s1 >= 0, s1 <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 2 }-> if(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   gen(z', z'', z1) -{ 2 }-> if(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z'' >= 0, z' - 1 >= 0
   gen(z', z'', z1) -{ 1 }-> if(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   generate(z', z'') -{ 1 }-> gen(z', z'', 0) :|: z' >= 0, z'' >= 0
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   if(z', z'', z1, z2) -{ 1 }-> 1 + z1 + gen(z'', z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> z2 :|: z'' >= 0, z1 >= 0, z' = 2, z2 >= 0
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(z'', z1, z2) :|: z'' >= 0, z1 >= 0, z2 >= 0, z' = 1
   ifsum2(z', z'', z1) -{ 2 }-> sum2(xs'', z1) :|: x1 >= 0, z' = 2, xs'' >= 0, z1 >= 0, z'' = 1 + x1 + xs''
   ifsum2(z', z'', z1) -{ 2 }-> sum2(0, z1) :|: z'' = 0, z' = 2, z1 >= 0
   ifsum2(z', z'', z1) -{ 5 + x2 }-> sum2(1 + s3 + xs1, 1 + z1) :|: s3 >= 0, s3 <= 2 + x2, z'' = 1 + x2 + xs1, z1 >= 0, xs1 >= 0, z' = 1, x2 >= 0
   ifsum2(z', z'', z1) -{ 6 }-> sum2(1 + s4 + 0, 1 + z1) :|: s4 >= 0, s4 <= 2 + 1, z'' = 0, z1 >= 0, z' = 1
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isZero(z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 2, z' - 2 >= 0
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: z' >= 0
   p(z') -{ 2 + z' }-> 1 + s5 :|: s5 >= 0, s5 <= 2 + (1 + (z' - 2)), z' - 2 >= 0
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(z', 0) :|: z' >= 0
   sum2(z', z'') -{ 6 }-> ifsum(2, s, 0, z'') :|: s >= 0, s <= 2, z'' >= 0, z' = 0
   sum2(z', z'') -{ 5 + x'' }-> ifsum(1, s', 1 + x'' + xs', z'') :|: s' >= 0, s' <= 2, xs' >= 0, z'' >= 0, z' = 1 + x'' + xs', x'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   times(z', z'') -{ 2 }-> sum(gen(z', z'', 0)) :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {ifsum2,ifsum,sum2}, {if,gen}, {sum}, {generate}, {times}
Previous analysis results are:
  isZero: runtime: O(n^1) [2 + z'], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  ge: runtime: O(n^1) [2 + z'], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  isNil: runtime: O(1) [1], size: O(1) [2]
  p: runtime: O(n^1) [2 + z'], size: O(n^1) [2 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  ifsum2: runtime: INF, size: INF
  ifsum: runtime: ?, size: INF
  sum2: runtime: ?, size: INF

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

(67) 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:

   isZero(v0) -> null_isZero [0]
   p(v0) -> null_p [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3) -> null_if [0]
   ifsum(v0, v1, v2, v3) -> null_ifsum [0]
   ifsum2(v0, v1, v2) -> null_ifsum2 [0]
   isNil(v0) -> null_isNil [0]
   tail(v0) -> null_tail [0]
   head(v0) -> null_head [0]

And the following fresh constants:    null_isZero, null_p, null_ge, null_if, null_ifsum, null_ifsum2, null_isNil, null_tail, null_head

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

(68)
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(xs) -> sum2(xs, 0) [1]
   sum2(xs, y) -> ifsum(isNil(xs), isZero(head(xs)), xs, y) [1]
   ifsum(true, b, xs, y) -> y [1]
   ifsum(false, b, xs, y) -> ifsum2(b, xs, y) [1]
   ifsum2(true, xs, y) -> sum2(tail(xs), y) [1]
   ifsum2(false, xs, y) -> sum2(cons(p(head(xs)), tail(xs)), s(y)) [1]
   isNil(nil) -> true [1]
   isNil(cons(x, xs)) -> false [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   head(cons(x, xs)) -> x [1]
   head(nil) -> error [1]
   isZero(0) -> true [1]
   isZero(s(0)) -> false [1]
   isZero(s(s(x))) -> isZero(s(x)) [1]
   p(0) -> s(s(0)) [1]
   p(s(0)) -> 0 [1]
   p(s(s(x))) -> s(p(s(x))) [1]
   ge(x, 0) -> true [1]
   ge(0, s(y)) -> false [1]
   ge(s(x), s(y)) -> ge(x, y) [1]
   a -> c [1]
   a -> d [1]
   isZero(v0) -> null_isZero [0]
   p(v0) -> null_p [0]
   ge(v0, v1) -> null_ge [0]
   if(v0, v1, v2, v3) -> null_if [0]
   ifsum(v0, v1, v2, v3) -> null_ifsum [0]
   ifsum2(v0, v1, v2) -> null_ifsum2 [0]
   isNil(v0) -> null_isNil [0]
   tail(v0) -> null_tail [0]
   head(v0) -> null_head [0]

The TRS has the following type information:
times :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
sum :: nil:cons:null_if:null_tail -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
generate :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> nil:cons:null_if:null_tail
gen :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> nil:cons:null_if:null_tail
0 :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
if :: true:false:null_isZero:null_ge:null_isNil -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> nil:cons:null_if:null_tail
ge :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> true:false:null_isZero:null_ge:null_isNil
true :: true:false:null_isZero:null_ge:null_isNil
nil :: nil:cons:null_if:null_tail
false :: true:false:null_isZero:null_ge:null_isNil
cons :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> nil:cons:null_if:null_tail -> nil:cons:null_if:null_tail
s :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
sum2 :: nil:cons:null_if:null_tail -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
ifsum :: true:false:null_isZero:null_ge:null_isNil -> true:false:null_isZero:null_ge:null_isNil -> nil:cons:null_if:null_tail -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
isNil :: nil:cons:null_if:null_tail -> true:false:null_isZero:null_ge:null_isNil
isZero :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> true:false:null_isZero:null_ge:null_isNil
head :: nil:cons:null_if:null_tail -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
ifsum2 :: true:false:null_isZero:null_ge:null_isNil -> nil:cons:null_if:null_tail -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
tail :: nil:cons:null_if:null_tail -> nil:cons:null_if:null_tail
p :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head -> 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
error :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
a :: c:d
c :: c:d
d :: c:d
null_isZero :: true:false:null_isZero:null_ge:null_isNil
null_p :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
null_ge :: true:false:null_isZero:null_ge:null_isNil
null_if :: nil:cons:null_if:null_tail
null_ifsum :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
null_ifsum2 :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head
null_isNil :: true:false:null_isZero:null_ge:null_isNil
null_tail :: nil:cons:null_if:null_tail
null_head :: 0:s:error:null_p:null_ifsum:null_ifsum2:null_head

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

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

   0 => 0
   true => 2
   nil => 0
   false => 1
   error => 1
   c => 0
   d => 1
   null_isZero => 0
   null_p => 0
   null_ge => 0
   null_if => 0
   null_ifsum => 0
   null_ifsum2 => 0
   null_isNil => 0
   null_tail => 0
   null_head => 0

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

(70)
Obligation:
Complexity RNTS consisting of the following rules:

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   ge(z', z'') -{ 1 }-> ge(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = x, x >= 0
   ge(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   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
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   head(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   if(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   if(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   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' = 1
   ifsum(z', z'', z1, z2) -{ 1 }-> y :|: b >= 0, xs >= 0, z2 = y, z'' = b, z' = 2, y >= 0, z1 = xs
   ifsum(z', z'', z1, z2) -{ 1 }-> ifsum2(b, xs, y) :|: b >= 0, xs >= 0, z2 = y, z'' = b, y >= 0, z' = 1, z1 = xs
   ifsum(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   ifsum2(z', z'', z1) -{ 1 }-> sum2(tail(xs), y) :|: xs >= 0, z1 = y, z' = 2, y >= 0, z'' = xs
   ifsum2(z', z'', z1) -{ 1 }-> sum2(1 + p(head(xs)) + tail(xs), 1 + y) :|: xs >= 0, z1 = y, y >= 0, z'' = xs, z' = 1
   ifsum2(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   isNil(z') -{ 1 }-> 2 :|: z' = 0
   isNil(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   isNil(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   isZero(z') -{ 1 }-> isZero(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   isZero(z') -{ 1 }-> 2 :|: z' = 0
   isZero(z') -{ 1 }-> 1 :|: z' = 1 + 0
   isZero(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 0 :|: z' = 1 + 0
   p(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   p(z') -{ 1 }-> 1 + p(1 + x) :|: x >= 0, z' = 1 + (1 + x)
   p(z') -{ 1 }-> 1 + (1 + 0) :|: z' = 0
   sum(z') -{ 1 }-> sum2(xs, 0) :|: xs >= 0, z' = xs
   sum2(z', z'') -{ 1 }-> ifsum(isNil(xs), isZero(head(xs)), xs, y) :|: xs >= 0, z'' = y, y >= 0, z' = xs
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0
   tail(z') -{ 0 }-> 0 :|: v0 >= 0, z' = v0
   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.

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

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

(72)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(true, z0, z1, z2) -> c4
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISNIL(z0))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c9
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1), TAIL(z0))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)), HEAD(z0))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), TAIL(z0))
   ISNIL(nil) -> c14
   ISNIL(cons(z0, z1)) -> c15
   TAIL(nil) -> c16
   TAIL(cons(z0, z1)) -> c17
   HEAD(cons(z0, z1)) -> c18
   HEAD(nil) -> c19
   ISZERO(0) -> c20
   ISZERO(s(0)) -> c21
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(0) -> c23
   P(s(0)) -> c24
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(z0, 0) -> c26
   GE(0, s(z0)) -> c27
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   A -> c29
   A -> c30
S tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(true, z0, z1, z2) -> c4
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISNIL(z0))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c9
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1), TAIL(z0))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)), HEAD(z0))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), TAIL(z0))
   ISNIL(nil) -> c14
   ISNIL(cons(z0, z1)) -> c15
   TAIL(nil) -> c16
   TAIL(cons(z0, z1)) -> c17
   HEAD(cons(z0, z1)) -> c18
   HEAD(nil) -> c19
   ISZERO(0) -> c20
   ISZERO(s(0)) -> c21
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(0) -> c23
   P(s(0)) -> c24
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(z0, 0) -> c26
   GE(0, s(z0)) -> c27
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   A -> c29
   A -> c30
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   TIMES_2, GENERATE_2, GEN_3, IF_4, SUM_1, SUM2_2, IFSUM_4, IFSUM2_3, ISNIL_1, TAIL_1, HEAD_1, ISZERO_1, P_1, GE_2, A

Compound Symbols:   c1_2, c2_1, c3_2, c4, c5_1, c6_1, c7_2, c8_3, c9, c10_1, c11_2, c12_3, c13_2, c14, c15, c16, c17, c18, c19, c20, c21, c22_1, c23, c24, c25_1, c26, c27, c28_1, c29, c30


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

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 16 trailing nodes:
   A -> c29
   IF(true, z0, z1, z2) -> c4
   TAIL(nil) -> c16
   IFSUM(true, z0, z1, z2) -> c9
   P(s(0)) -> c24
   P(0) -> c23
   TAIL(cons(z0, z1)) -> c17
   GE(z0, 0) -> c26
   HEAD(cons(z0, z1)) -> c18
   GE(0, s(z0)) -> c27
   HEAD(nil) -> c19
   ISNIL(nil) -> c14
   A -> c30
   ISZERO(s(0)) -> c21
   ISNIL(cons(z0, z1)) -> c15
   ISZERO(0) -> c20

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

(74)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISNIL(z0))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)), HEAD(z0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1), TAIL(z0))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)), HEAD(z0))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), TAIL(z0))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
S tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISNIL(z0))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)), HEAD(z0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1), TAIL(z0))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)), HEAD(z0))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), TAIL(z0))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   TIMES_2, GENERATE_2, GEN_3, IF_4, SUM_1, SUM2_2, IFSUM_4, IFSUM2_3, ISZERO_1, P_1, GE_2

Compound Symbols:   c1_2, c2_1, c3_2, c5_1, c6_1, c7_2, c8_3, c10_1, c11_2, c12_3, c13_2, c22_1, c25_1, c28_1


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

(75) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing tuple parts
----------------------------------------

(76)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
S tuples:
   TIMES(z0, z1) -> c1(SUM(generate(z0, z1)), GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   TIMES_2, GENERATE_2, GEN_3, IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3

Compound Symbols:   c1_2, c2_1, c3_2, c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1


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

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

(78)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   TIMES(z0, z1) -> c4(GENERATE(z0, z1))
S tuples:
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   TIMES(z0, z1) -> c4(GENERATE(z0, z1))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   GENERATE_2, GEN_3, IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2

Compound Symbols:   c2_1, c3_2, c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1


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

(79) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   TIMES(z0, z1) -> c4(GENERATE(z0, z1))
   GENERATE(z0, z1) -> c2(GEN(z0, z1, 0))

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

(80)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
K tuples:none
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2

Compound Symbols:   c3_2, c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1


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

(81) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))

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

(82)
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(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   a -> c
   a -> d
Tuples:
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   times_2, generate_2, gen_3, if_4, sum_1, sum2_2, ifsum_4, ifsum2_3, isNil_1, tail_1, head_1, isZero_1, p_1, ge_2, a

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2

Compound Symbols:   c3_2, c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1


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

(83) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   times(z0, z1) -> sum(generate(z0, z1))
   sum(z0) -> sum2(z0, 0)
   sum2(z0, z1) -> ifsum(isNil(z0), isZero(head(z0)), z0, z1)
   ifsum(true, z0, z1, z2) -> z2
   ifsum(false, z0, z1, z2) -> ifsum2(z0, z1, z2)
   ifsum2(true, z0, z1) -> sum2(tail(z0), z1)
   ifsum2(false, z0, z1) -> sum2(cons(p(head(z0)), tail(z0)), s(z1))
   a -> c
   a -> d

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
S tuples:
   GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0))
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   GEN_3, IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2

Compound Symbols:   c3_2, c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(z0, z1, z2) -> c3(IF(ge(z2, z0), z0, z1, z2), GE(z2, z0)) by 
   GEN(0, x1, z0) -> c3(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c3(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(0, x1, z0) -> c3(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c3(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(0, x1, z0) -> c3(IF(true, 0, x1, z0), GE(z0, 0))
   GEN(s(z0), x1, 0) -> c3(IF(false, s(z0), x1, 0), GE(0, s(z0)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1, c3_2


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

(87) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   GEN(s(z0), x1, 0) -> c3(IF(false, s(z0), x1, 0), GE(0, s(z0)))
Removed 1 trailing nodes:
   GEN(0, x1, z0) -> c3(IF(true, 0, x1, z0), GE(z0, 0))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c7_1, c8_2, c11_1, c12_2, c13_1, c4_1, c3_2


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(z0, z1) -> c7(IFSUM(isNil(z0), isZero(head(z0)), z0, z1)) by 
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c7(IFSUM(true, isZero(head(nil)), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c7(IFSUM(true, isZero(head(nil)), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c7(IFSUM(true, isZero(head(nil)), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c8_2, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1


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

(91) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUM2(nil, x1) -> c7(IFSUM(true, isZero(head(nil)), nil, x1))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0)))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, SUM2_2, IFSUM2_3, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c8_2, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1


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

(93) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(z0, z1) -> c8(IFSUM(isNil(z0), isZero(head(z0)), z0, z1), ISZERO(head(z0))) by 
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(true, isZero(head(nil)), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(true, isZero(head(nil)), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(true, isZero(head(nil)), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1, c8_2


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

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1


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

(97) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   generate(z0, z1) -> gen(z0, z1, 0)
   isNil(cons(z0, z1)) -> false
   gen(z0, z1, z2) -> if(ge(z2, z0), z0, z1, z2)
   if(false, z0, z1, z2) -> cons(z1, gen(z0, z1, s(z2)))
   if(true, z0, z1, z2) -> nil
   tail(nil) -> nil
   isNil(nil) -> true
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
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)) = x_1 + x_3
   POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4
   POL(IFSUM2(x_1, x_2, x_3)) = x_2 + x_3
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = x_1
   POL(SUM2(x_1, x_2)) = x_1 + x_2
   POL(TIMES(x_1, x_2)) = [1] + x_1 + x_2
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = x_2
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = [1] + x_2
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2
   POL(generate(x_1, x_2)) = [1] + x_1 + x_2
   POL(head(x_1)) = 0
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3
   POL(isNil(x_1)) = 0
   POL(isZero(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1


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

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
We considered the (Usable) Rules:
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
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)) = x_1 + x_3
   POL(IF(x_1, x_2, x_3, x_4)) = x_2 + x_4
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(IFSUM2(x_1, x_2, x_3)) = [1] + x_3
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = [1]
   POL(SUM2(x_1, x_2)) = [1] + x_2
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = [1]
   POL(error) = [1]
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1] + x_1 + x_2
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(generate(x_1, x_2)) = [1] + x_1 + x_2
   POL(head(x_1)) = 0
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(isNil(x_1)) = x_1
   POL(isZero(x_1)) = [1]
   POL(nil) = 0
   POL(p(x_1)) = [1] + x_1
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c11_1, c12_2, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(true, z0, z1) -> c11(SUM2(tail(z0), z1)) by 
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0)))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c12_2, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1, c11_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, z0, z1) -> c12(SUM2(cons(p(head(z0)), tail(z0)), s(z1)), P(head(z0))) by 
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)), P(error))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)), P(error))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)), P(error))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1


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

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c4_1, c3_2, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c4(SUM(generate(z0, z1))) by 
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(nil) -> true
   isNil(cons(z0, z1)) -> false
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   ge_2, isNil_1, isZero_1, head_1, tail_1, p_1, generate_2, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c3_2, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1


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

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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c3_2, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(s(z1), x1, s(z0)) -> c3(IF(ge(z0, z1), s(z1), x1, s(z0)), GE(s(z0), s(z1))) by 
   GEN(s(0), x1, s(z0)) -> c3(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(0), x1, s(z0)) -> c3(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(0), x1, s(z0)) -> c3(IF(true, s(0), x1, s(z0)), GE(s(z0), s(0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(115) 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)) -> c3(GE(s(z0), s(0)))
We considered the (Usable) Rules:
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(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(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_4
   POL(IFSUM2(x_1, x_2, x_3)) = x_3
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = [1]
   POL(SUM2(x_1, x_2)) = x_2
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = 0
   POL(false) = 0
   POL(ge(x_1, x_2)) = [1]
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(head(x_1)) = x_1
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(isNil(x_1)) = 0
   POL(isZero(x_1)) = [1] + x_1
   POL(nil) = 0
   POL(p(x_1)) = [1]
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(117) 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)) -> c3(GE(s(x2), s(x0)))
We considered the (Usable) Rules:
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), 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)) = x_1
   POL(IF(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1
   POL(IFSUM2(x_1, x_2, x_3)) = 0
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = [1]
   POL(SUM2(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = [1]
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = 0
   POL(false) = 0
   POL(ge(x_1, x_2)) = [1]
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(head(x_1)) = x_1
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(isNil(x_1)) = 0
   POL(isZero(x_1)) = [1] + x_1
   POL(nil) = 0
   POL(p(x_1)) = [1]
   POL(s(x_1)) = [1]
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x1) -> c7(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1)) by 
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(121) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1)) by 
   SUM2(nil, x0) -> c7(IFSUM(true, isZero(error), nil, x0))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(nil, x0) -> c7(IFSUM(true, isZero(error), nil, x0))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
K tuples:
   TIMES(z0, z1) -> c4(SUM(generate(z0, z1)))
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c7(IFSUM(isNil(nil), isZero(error), nil, x1))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(123) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUM2(nil, x0) -> c7(IFSUM(true, isZero(error), nil, x0))

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c7_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1


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

(125) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x1) -> c7(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1)) by 
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

(127) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x1) -> c8(IFSUM(isNil(cons(z0, z1)), isZero(z0), cons(z0, z1), x1), ISZERO(head(cons(z0, z1)))) by 
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

(129) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil))) by 
   SUM2(nil, x0) -> c8(IFSUM(true, isZero(error), nil, x0), ISZERO(head(nil)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x0) -> c8(IFSUM(true, isZero(error), nil, x0), ISZERO(head(nil)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

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

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
   isNil(nil) -> true
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, ge_2, isNil_1, isZero_1, gen_3, if_4

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

(133) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   isNil(nil) -> true

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

(135) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
We considered the (Usable) Rules:
   isNil(cons(z0, z1)) -> false
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
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)) = x_1 + x_3
   POL(IF(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_3 + x_4
   POL(IFSUM2(x_1, x_2, x_3)) = x_2 + x_3
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = 0
   POL(SUM2(x_1, x_2)) = x_2
   POL(TIMES(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1 + x_2
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(head(x_1)) = [1]
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(isNil(x_1)) = 0
   POL(isZero(x_1)) = [1]
   POL(nil) = [1]
   POL(p(x_1)) = [1]
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0

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

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1))))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_2, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1


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

(137) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x1) -> c8(IFSUM(false, isZero(head(cons(z0, z1))), cons(z0, z1), x1), ISZERO(head(cons(z0, z1)))) by 
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, SUM2_2, TIMES_2, GEN_3

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c8_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1, c8_2


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

(139) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(nil, x1) -> c8(ISZERO(head(nil))) by 
   SUM2(nil, x0) -> c8(ISZERO(error))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   SUM2(nil, x0) -> c8(ISZERO(error))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   SUM2(nil, x1) -> c8(ISZERO(head(nil)))
   SUM2(nil, x1) -> c8(IFSUM(isNil(nil), isZero(error), nil, x1), ISZERO(head(nil)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1, c8_2, c8_1


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

(141) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   SUM2(nil, x0) -> c8(ISZERO(error))
   IFSUM2(true, nil, x1) -> c11(SUM2(nil, x1))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_2, c12_1, c4_1, c3_2, c3_1, c7_1, c8_2


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, cons(z0, z1), x1) -> c12(SUM2(cons(p(head(cons(z0, z1))), tail(cons(z0, z1))), s(x1)), P(z0)) by 
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c4_1, c3_2, c3_1, c7_1, c8_2, c12_2


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

(145) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1))) by 
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, TIMES_2, GEN_3, SUM2_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c4_1, c3_2, c3_1, c7_1, c8_2, c12_2


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

(147) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c4(SUM(gen(z0, z1, 0))) by 
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(s(z1)), x1, s(s(z0))) -> c3(IF(ge(z0, z1), s(s(z1)), x1, s(s(z0))), GE(s(s(z0)), s(s(z1))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(149) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(s(s(z1)), x1, s(s(z0))) -> c3(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))) -> c3(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))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))

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

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(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))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(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))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(153) 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(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
We considered the (Usable) Rules:none
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(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(IFSUM(x_1, x_2, x_3, x_4)) = 0
   POL(IFSUM2(x_1, x_2, x_3)) = 0
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = 0
   POL(SUM2(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = [1] + x_1
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = 0
   POL(false) = 0
   POL(ge(x_1, x_2)) = 0
   POL(gen(x_1, x_2, x_3)) = [3] + [3]x_1 + [3]x_2 + [3]x_3
   POL(head(x_1)) = 0
   POL(if(x_1, x_2, x_3, x_4)) = [2]x_2 + [3]x_3
   POL(isNil(x_1)) = [3] + [3]x_1
   POL(isZero(x_1)) = [3]
   POL(nil) = 0
   POL(p(x_1)) = [3]
   POL(s(x_1)) = x_1
   POL(tail(x_1)) = [3]
   POL(true) = 0

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(155) 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))) -> c3(GE(s(s(x2)), s(s(x0))))
We considered the (Usable) Rules:
   isNil(cons(z0, z1)) -> false
And the Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(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)) = x_1
   POL(IF(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1
   POL(IFSUM2(x_1, x_2, x_3)) = 0
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = 0
   POL(SUM2(x_1, x_2)) = 0
   POL(TIMES(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = [1]
   POL(false) = 0
   POL(ge(x_1, x_2)) = x_1 + x_2
   POL(gen(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(head(x_1)) = [1] + x_1
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(isNil(x_1)) = 0
   POL(isZero(x_1)) = [1]
   POL(nil) = 0
   POL(p(x_1)) = [1]
   POL(s(x_1)) = [1] + x_1
   POL(tail(x_1)) = [1] + x_1
   POL(true) = 0

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(157) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(0, x1), x2) -> c7(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2)) by 
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(159) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(0), x1), x2) -> c7(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2)) by 
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(161) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(z0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2)) by 
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))

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

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1))))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(163) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(0, x1), x2) -> c8(IFSUM(isNil(cons(0, x1)), true, cons(0, x1), x2), ISZERO(head(cons(0, x1)))) by 
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1))))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(165) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(0), x1), x2) -> c8(IFSUM(isNil(cons(s(0), x1)), false, cons(s(0), x1), x2), ISZERO(head(cons(s(0), x1)))) by 
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(167) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(z0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(z0)), x1)), isZero(s(z0)), cons(s(s(z0)), x1), x2), ISZERO(head(cons(s(s(z0)), x1)))) by 
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1


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

(169) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1)))) by 
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1))))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c8_2, c12_2, c4_1, c8_1


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(head(cons(z0, z1)))) by 
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c12_2, c4_1, c8_2, c8_1


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

(173) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(head(cons(z0, z1))), z1), s(x2)), P(z0)) by 
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c12_2, c4_1, c8_2, c8_1


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

(175) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), tail(cons(z0, z1))), s(x2)), P(z0)) by 
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)), P(0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)), P(s(0)))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)), P(0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)), P(s(0)))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)), P(0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)), P(s(0)))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(177) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing tuple parts
----------------------------------------

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(179) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(head(nil)), nil), s(x0))) by 
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0)))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(181) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), tail(nil)), s(x0))) by 
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(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))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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)))) -> c3(IF(ge(z0, z1), s(s(s(z1))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(z1)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(183) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace GEN(s(s(s(z1))), x1, s(s(s(z0)))) -> c3(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(0))), x1, s(s(s(z0)))) -> c3(IF(true, s(s(s(0))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(IF(true, s(s(s(0))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(IF(true, s(s(s(0))), x1, s(s(s(z0)))), GE(s(s(s(z0))), s(s(s(0)))))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(187) 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(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(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) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(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(IFSUM(x_1, x_2, x_3, x_4)) = x_4
   POL(IFSUM2(x_1, x_2, x_3)) = x_3
   POL(ISZERO(x_1)) = 0
   POL(P(x_1)) = 0
   POL(SUM(x_1)) = 0
   POL(SUM2(x_1, x_2)) = x_2
   POL(TIMES(x_1, x_2)) = 0
   POL(c10(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c12(x_1)) = x_1
   POL(c12(x_1, x_2)) = x_1 + x_2
   POL(c13(x_1)) = x_1
   POL(c22(x_1)) = x_1
   POL(c25(x_1)) = x_1
   POL(c28(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(cons(x_1, x_2)) = 0
   POL(error) = [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(head(x_1)) = [1]
   POL(if(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(isNil(x_1)) = [1] + x_1
   POL(isZero(x_1)) = [1]
   POL(nil) = 0
   POL(p(x_1)) = [1]
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1]
   POL(true) = 0

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(189) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(0)), x1), x2) -> c7(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2)) by 
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))

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

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(s(z0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2)) by 
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1))))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(193) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(head(cons(0, z1)))) by 
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(0))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(0))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, z1), x1) -> c8(IFSUM(false, true, cons(0, z1), x1), ISZERO(0))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

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

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1))))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(197) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(head(cons(s(0), z1)))) by 
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(s(0)))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(s(0)))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), z1), x1) -> c8(IFSUM(false, false, cons(s(0), z1), x1), ISZERO(s(0)))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

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

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1))))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(201) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(0)), x1), x2) -> c8(IFSUM(isNil(cons(s(s(0)), x1)), false, cons(s(s(0)), x1), x2), ISZERO(head(cons(s(s(0)), x1)))) by 
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(203) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(s(z0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(z0))), x1)), isZero(s(z0)), cons(s(s(s(z0))), x1), x2), ISZERO(head(cons(s(s(s(z0))), x1)))) by 
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1))))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(205) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(head(cons(s(s(x0)), z1)))) by 
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))

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

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
S tuples:
   IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2)))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   IF_4, SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2

Compound Symbols:   c5_1, c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2


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

(207) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, z0, z1, z2) -> c5(GEN(z0, z1, s(z2))) by 
   IF(false, s(s(x0)), x1, s(0)) -> c5(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c5(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(0)))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(0))))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(x0)), x1, s(0)) -> c5(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c5(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(0)))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(0))))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
S tuples:
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(IF(false, s(s(s(z0))), x1, s(s(0))), GE(s(s(0)), s(s(s(z0)))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(x0)), x1, s(0)) -> c5(GEN(s(s(x0)), x1, s(s(0))))
   IF(false, s(s(s(x0))), x1, s(s(0))) -> c5(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(0)))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(0))))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   IFSUM2(false, nil, x1) -> c12(SUM2(cons(p(head(nil)), tail(nil)), s(x1)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4

Compound Symbols:   c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_2, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c5_1


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

(209) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 9 leading nodes:
   GEN(s(s(z0)), x1, s(0)) -> c3(IF(false, s(s(z0)), x1, s(0)), GE(s(0), s(s(z0))))
   IF(false, s(s(x0)), x1, s(0)) -> c5(GEN(s(s(x0)), x1, s(s(0))))
   GEN(s(s(s(z0))), x1, s(s(0))) -> c3(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))) -> c5(GEN(s(s(s(x0))), x1, s(s(s(0)))))
   GEN(s(s(s(s(z0)))), x1, s(s(s(0)))) -> c3(IF(false, s(s(s(s(z0)))), x1, s(s(s(0)))), GE(s(s(s(0))), s(s(s(s(z0))))))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(0)))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(0))))))
   GEN(s(0), x1, s(z0)) -> c3(GE(s(z0), s(0)))
   GEN(s(s(0)), x1, s(s(z0))) -> c3(GE(s(s(z0)), s(s(0))))
   GEN(s(s(s(0))), x1, s(s(s(z0)))) -> c3(GE(s(s(s(z0))), s(s(s(0)))))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
S tuples:
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0))))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4

Compound Symbols:   c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1


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

(211) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFSUM2(false, cons(s(s(z0)), x1), x2) -> c12(SUM2(cons(s(p(s(z0))), tail(cons(s(s(z0)), x1))), s(x2)), P(s(s(z0)))) by IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
----------------------------------------

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
S tuples:
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2)))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4

Compound Symbols:   c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1


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

(213) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFSUM2(false, cons(0, x1), x2) -> c12(SUM2(cons(s(s(0)), tail(cons(0, x1))), s(x2))) by IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
----------------------------------------

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
S tuples:
   IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, IFSUM_4, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4

Compound Symbols:   c6_1, c10_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1


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

(215) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFSUM(false, z0, z1, z2) -> c10(IFSUM2(z0, z1, z2)) by 
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))

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

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4, IFSUM_4

Compound Symbols:   c6_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1


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

(217) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace IFSUM2(false, cons(s(0), x1), x2) -> c12(SUM2(cons(0, tail(cons(s(0), x1))), s(x2))) by IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
----------------------------------------

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
K tuples:
   SUM(z0) -> c6(SUM2(z0, 0))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   SUM_1, ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4, IFSUM_4

Compound Symbols:   c6_1, c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1


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

(219) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(z0) -> c6(SUM2(z0, 0)) by 
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))

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

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
K tuples:
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, TIMES_2, IF_4, IFSUM_4, SUM_1

Compound Symbols:   c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c4_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1, c6_1


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

(221) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace TIMES(z0, z1) -> c4(SUM(if(ge(0, z0), z0, z1, 0))) by 
   TIMES(0, x1) -> c4(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c4(SUM(if(false, s(z0), x1, 0)))

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

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
   TIMES(0, x1) -> c4(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c4(SUM(if(false, s(z0), x1, 0)))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
K tuples:
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, IF_4, IFSUM_4, SUM_1, TIMES_2

Compound Symbols:   c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1, c6_1, c4_1


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

(223) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUM2(cons(s(s(s(0))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2)) by SUM2(cons(s(s(s(0))), z0), z1) -> c7(IFSUM(false, false, cons(s(s(s(0))), z0), z1))
----------------------------------------

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
   TIMES(0, x1) -> c4(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c4(SUM(if(false, s(z0), x1, 0)))
   SUM2(cons(s(s(s(0))), z0), z1) -> c7(IFSUM(false, false, cons(s(s(s(0))), z0), z1))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM2(cons(s(s(s(0))), z0), z1) -> c7(IFSUM(false, false, cons(s(s(s(0))), z0), z1))
K tuples:
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, IF_4, IFSUM_4, SUM_1, TIMES_2

Compound Symbols:   c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1, c6_1, c4_1


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

(225) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c7(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2)) by SUM2(cons(s(s(s(s(z0)))), z1), z2) -> c7(IFSUM(false, isZero(s(z0)), cons(s(s(s(s(z0)))), z1), z2))
----------------------------------------

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   p(0) -> s(s(0))
   p(s(0)) -> 0
   p(s(s(z0))) -> s(p(s(z0)))
   head(cons(z0, z1)) -> z0
   head(nil) -> error
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   isZero(0) -> true
   isZero(s(0)) -> false
   isZero(s(s(z0))) -> isZero(s(z0))
   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)))
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   isNil(cons(z0, z1)) -> false
Tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   IFSUM2(false, nil, x0) -> c12(SUM2(cons(p(error), nil), s(x0)))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
   TIMES(0, x1) -> c4(SUM(if(true, 0, x1, 0)))
   TIMES(s(z0), x1) -> c4(SUM(if(false, s(z0), x1, 0)))
   SUM2(cons(s(s(s(0))), z0), z1) -> c7(IFSUM(false, false, cons(s(s(s(0))), z0), z1))
   SUM2(cons(s(s(s(s(z0)))), z1), z2) -> c7(IFSUM(false, isZero(s(z0)), cons(s(s(s(s(z0)))), z1), z2))
S tuples:
   ISZERO(s(s(z0))) -> c22(ISZERO(s(z0)))
   P(s(s(z0))) -> c25(P(s(z0)))
   GE(s(z0), s(z1)) -> c28(GE(z0, z1))
   IFSUM2(false, z0, z1) -> c13(SUM2(cons(p(head(z0)), tail(z0)), s(z1)))
   IFSUM2(true, cons(z0, z1), x1) -> c11(SUM2(z1, x1))
   IFSUM2(false, x0, x1) -> c12(SUM2(cons(p(head(x0)), tail(x0)), s(x1)))
   SUM2(cons(z0, z1), x2) -> c7(IFSUM(false, isZero(z0), cons(z0, z1), x2))
   SUM2(cons(0, z1), x1) -> c7(IFSUM(false, true, cons(0, z1), x1))
   SUM2(cons(s(0), z1), x1) -> c7(IFSUM(false, false, cons(s(0), z1), x1))
   SUM2(cons(s(s(x0)), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2))
   SUM2(cons(z0, z1), x2) -> c8(IFSUM(false, isZero(z0), cons(z0, z1), x2), ISZERO(z0))
   SUM2(cons(x0, x1), x2) -> c8(IFSUM(false, isZero(x0), cons(x0, x1), x2))
   IFSUM2(false, cons(z0, z1), x2) -> c12(SUM2(cons(p(z0), z1), s(x2)), P(z0))
   GEN(s(s(s(s(z1)))), x1, s(s(s(s(z0))))) -> c3(IF(ge(z0, z1), s(s(s(s(z1)))), x1, s(s(s(s(z0))))), GE(s(s(s(s(z0)))), s(s(s(s(z1))))))
   SUM2(cons(s(s(0)), z1), x1) -> c7(IFSUM(false, false, cons(s(s(0)), z1), x1))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c7(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2))
   SUM2(cons(0, x0), x1) -> c8(IFSUM(false, true, cons(0, x0), x1))
   SUM2(cons(s(0), x0), x1) -> c8(IFSUM(false, false, cons(s(0), x0), x1))
   SUM2(cons(s(s(0)), z1), x1) -> c8(IFSUM(false, false, cons(s(s(0)), z1), x1), ISZERO(head(cons(s(s(0)), z1))))
   SUM2(cons(s(s(s(0))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(0))), x1)), false, cons(s(s(s(0))), x1), x2), ISZERO(head(cons(s(s(s(0))), x1))))
   SUM2(cons(s(s(s(s(z0)))), x1), x2) -> c8(IFSUM(isNil(cons(s(s(s(s(z0)))), x1)), isZero(s(z0)), cons(s(s(s(s(z0)))), x1), x2), ISZERO(head(cons(s(s(s(s(z0)))), x1))))
   SUM2(cons(s(s(s(x0))), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(s(x0))), z1), x2), ISZERO(head(cons(s(s(s(x0))), z1))))
   SUM2(cons(s(s(x0)), z1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), z1), x2), ISZERO(s(s(x0))))
   SUM2(cons(s(s(x0)), x1), x2) -> c8(IFSUM(false, isZero(s(x0)), cons(s(s(x0)), x1), x2))
   IF(false, s(s(s(s(x0)))), x1, s(s(s(s(x2))))) -> c5(GEN(s(s(s(s(x0)))), x1, s(s(s(s(s(x2)))))))
   IFSUM2(false, cons(s(s(z0)), z1), z2) -> c12(SUM2(cons(s(p(s(z0))), z1), s(z2)), P(s(s(z0))))
   IFSUM2(false, cons(0, z0), z1) -> c12(SUM2(cons(s(s(0)), z0), s(z1)))
   IFSUM(false, y0, cons(x0, x1), x2) -> c10(IFSUM2(y0, cons(x0, x1), x2))
   IFSUM(false, true, cons(0, x0), x1) -> c10(IFSUM2(true, cons(0, x0), x1))
   IFSUM(false, false, cons(s(0), x0), x1) -> c10(IFSUM2(false, cons(s(0), x0), x1))
   IFSUM(false, y0, cons(s(s(x0)), x1), x2) -> c10(IFSUM2(y0, cons(s(s(x0)), x1), x2))
   IFSUM(false, false, cons(s(s(0)), x0), x1) -> c10(IFSUM2(false, cons(s(s(0)), x0), x1))
   IFSUM(false, false, cons(s(s(s(0))), x0), x1) -> c10(IFSUM2(false, cons(s(s(s(0))), x0), x1))
   IFSUM(false, y1, cons(s(s(s(s(x0)))), x1), x2) -> c10(IFSUM2(y1, cons(s(s(s(s(x0)))), x1), x2))
   IFSUM(false, y0, cons(s(s(s(x0))), x1), x2) -> c10(IFSUM2(y0, cons(s(s(s(x0))), x1), x2))
   IFSUM2(false, cons(s(0), z0), z1) -> c12(SUM2(cons(0, z0), s(z1)))
   SUM2(cons(s(s(s(0))), z0), z1) -> c7(IFSUM(false, false, cons(s(s(s(0))), z0), z1))
   SUM2(cons(s(s(s(s(z0)))), z1), z2) -> c7(IFSUM(false, isZero(s(z0)), cons(s(s(s(s(z0)))), z1), z2))
K tuples:
   GEN(s(x0), x1, s(x2)) -> c3(GE(s(x2), s(x0)))
   GEN(s(s(x0)), x1, s(s(x2))) -> c3(GE(s(s(x2)), s(s(x0))))
   GEN(s(s(s(x0))), x1, s(s(s(x2)))) -> c3(GE(s(s(s(x2))), s(s(s(x0)))))
   SUM(cons(y0, y1)) -> c6(SUM2(cons(y0, y1), 0))
   SUM(cons(0, y0)) -> c6(SUM2(cons(0, y0), 0))
   SUM(cons(s(0), y0)) -> c6(SUM2(cons(s(0), y0), 0))
   SUM(cons(s(s(y0)), y1)) -> c6(SUM2(cons(s(s(y0)), y1), 0))
   SUM(cons(s(s(0)), y0)) -> c6(SUM2(cons(s(s(0)), y0), 0))
   SUM(cons(s(s(s(0))), y0)) -> c6(SUM2(cons(s(s(s(0))), y0), 0))
   SUM(cons(s(s(s(s(y0)))), y1)) -> c6(SUM2(cons(s(s(s(s(y0)))), y1), 0))
   SUM(cons(s(s(s(y0))), y1)) -> c6(SUM2(cons(s(s(s(y0))), y1), 0))
Defined Rule Symbols:   p_1, head_1, tail_1, isZero_1, gen_3, if_4, ge_2, isNil_1

Defined Pair Symbols:   ISZERO_1, P_1, GE_2, IFSUM2_3, GEN_3, SUM2_2, IF_4, IFSUM_4, SUM_1, TIMES_2

Compound Symbols:   c22_1, c25_1, c28_1, c13_1, c11_1, c12_1, c3_1, c7_1, c8_2, c8_1, c12_2, c3_2, c5_1, c10_1, c6_1, c4_1