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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 355 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 163 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 51 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 6 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 32 ms]
        (48) CpxRNTS
        (49) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (50) CpxRNTS
        (51) IntTrsBoundProof [UPPER BOUND(ID), 4068 ms]
        (52) CpxRNTS
        (53) IntTrsBoundProof [UPPER BOUND(ID), 928 ms]
        (54) CpxRNTS
    (55) CompletionProof [UPPER BOUND(ID), 0 ms]
    (56) CpxTypedWeightedCompleteTrs
        (57) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (58) CpxRNTS
(59) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(60) CdtProblem
    (61) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 37 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 41 ms]
    (106) CdtProblem
    (107) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 36 ms]
    (116) CdtProblem
    (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 1 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem
    (171) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (174) CdtProblem
    (175) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (176) CdtProblem
    (177) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (182) CdtProblem
    (183) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (192) CdtProblem
    (193) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) 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:

   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   sum(xs) -> sumIter(xs, 0)
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
   ifSum(true, xs, x, y) -> x
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   a -> b
   a -> c

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:

   plus(x, y) -> plusIter(x, y, 0')
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   le(s(x), 0') -> false
   le(0', y) -> true
   le(s(x), s(y)) -> le(x, y)
   sum(xs) -> sumIter(xs, 0')
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
   ifSum(true, xs, x, y) -> x
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   a -> b
   a -> c

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:

   plus(x, y) -> plusIter(x, y, 0)
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z)
   ifPlus(true, x, y, z) -> y
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z))
   le(s(x), 0) -> false
   le(0, y) -> true
   le(s(x), s(y)) -> le(x, y)
   sum(xs) -> sumIter(xs, 0)
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs)))
   ifSum(true, xs, x, y) -> x
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y)
   isempty(nil) -> true
   isempty(cons(x, xs)) -> false
   head(nil) -> error
   head(cons(x, xs)) -> x
   tail(nil) -> nil
   tail(cons(x, xs)) -> xs
   a -> b
   a -> c

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:

   plus(x, y) -> plusIter(x, y, 0) [1]
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1]
   ifPlus(true, x, y, z) -> y [1]
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1]
   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   sum(xs) -> sumIter(xs, 0) [1]
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1]
   ifSum(true, xs, x, y) -> x [1]
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   a -> b [1]
   a -> c [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:

   plus(x, y) -> plusIter(x, y, 0) [1]
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1]
   ifPlus(true, x, y, z) -> y [1]
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1]
   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   sum(xs) -> sumIter(xs, 0) [1]
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1]
   ifSum(true, xs, x, y) -> x [1]
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   a -> b [1]
   a -> c [1]

The TRS has the following type information:
plus :: 0:s:error -> 0:s:error -> 0:s:error
plusIter :: 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error
0 :: 0:s:error
ifPlus :: true:false -> 0:s:error -> 0:s:error -> 0:s:error -> 0:s:error
le :: 0:s:error -> 0:s:error -> true:false
true :: true:false
false :: true:false
s :: 0:s:error -> 0:s:error
sum :: nil:cons -> 0:s:error
sumIter :: nil:cons -> 0:s:error -> 0:s:error
ifSum :: true:false -> nil:cons -> 0:s:error -> 0:s:error -> 0:s:error
isempty :: nil:cons -> true:false
head :: nil:cons -> 0:s:error
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s:error -> nil:cons -> nil:cons
error :: 0:s:error
a :: b:c
b :: b:c
c :: b:c

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:

   sum_1
   sumIter_2
   ifSum_4
   a

(c) The following functions are completely defined:

   le_2
   isempty_1
   plus_2
   head_1
   tail_1
   plusIter_3
   ifPlus_4

Due to the following rules being added:

   le(v0, v1) -> null_le [0]
   ifPlus(v0, v1, v2, v3) -> null_ifPlus [0]

And the following fresh constants:    null_le, null_ifPlus

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

(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:

   plus(x, y) -> plusIter(x, y, 0) [1]
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1]
   ifPlus(true, x, y, z) -> y [1]
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1]
   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   sum(xs) -> sumIter(xs, 0) [1]
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1]
   ifSum(true, xs, x, y) -> x [1]
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   a -> b [1]
   a -> c [1]
   le(v0, v1) -> null_le [0]
   ifPlus(v0, v1, v2, v3) -> null_ifPlus [0]

The TRS has the following type information:
plus :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
plusIter :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
0 :: 0:s:error:null_ifPlus
ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
le :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> true:false:null_le
true :: true:false:null_le
false :: true:false:null_le
s :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
sum :: nil:cons -> 0:s:error:null_ifPlus
sumIter :: nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
isempty :: nil:cons -> true:false:null_le
head :: nil:cons -> 0:s:error:null_ifPlus
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s:error:null_ifPlus -> nil:cons -> nil:cons
error :: 0:s:error:null_ifPlus
a :: b:c
b :: b:c
c :: b:c
null_le :: true:false:null_le
null_ifPlus :: 0:s:error:null_ifPlus

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:

   plus(x, y) -> plusIter(x, y, 0) [1]
   plusIter(s(x'), y, 0) -> ifPlus(false, s(x'), y, 0) [2]
   plusIter(0, y, z) -> ifPlus(true, 0, y, z) [2]
   plusIter(s(x''), y, s(y')) -> ifPlus(le(x'', y'), s(x''), y, s(y')) [2]
   plusIter(x, y, z) -> ifPlus(null_le, x, y, z) [1]
   ifPlus(true, x, y, z) -> y [1]
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1]
   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   sum(xs) -> sumIter(xs, 0) [1]
   sumIter(nil, x) -> ifSum(true, nil, x, plus(x, error)) [3]
   sumIter(cons(x1, xs'), x) -> ifSum(false, cons(x1, xs'), x, plus(x, x1)) [3]
   ifSum(true, xs, x, y) -> x [1]
   ifSum(false, nil, x, y) -> sumIter(nil, y) [2]
   ifSum(false, cons(x2, xs''), x, y) -> sumIter(xs'', y) [2]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   a -> b [1]
   a -> c [1]
   le(v0, v1) -> null_le [0]
   ifPlus(v0, v1, v2, v3) -> null_ifPlus [0]

The TRS has the following type information:
plus :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
plusIter :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
0 :: 0:s:error:null_ifPlus
ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
le :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> true:false:null_le
true :: true:false:null_le
false :: true:false:null_le
s :: 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
sum :: nil:cons -> 0:s:error:null_ifPlus
sumIter :: nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus -> 0:s:error:null_ifPlus
isempty :: nil:cons -> true:false:null_le
head :: nil:cons -> 0:s:error:null_ifPlus
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s:error:null_ifPlus -> nil:cons -> nil:cons
error :: 0:s:error:null_ifPlus
a :: b:c
b :: b:c
c :: b:c
null_le :: true:false:null_le
null_ifPlus :: 0:s:error:null_ifPlus

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

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

   0 => 0
   true => 2
   false => 1
   nil => 0
   error => 1
   b => 0
   c => 1
   null_le => 0
   null_ifPlus => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(x, 1 + y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   ifSum(z', z'', z1, z2) -{ 1 }-> x :|: xs >= 0, z2 = y, z' = 2, x >= 0, y >= 0, z'' = xs, z1 = x
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', y) :|: z2 = y, z'' = 1 + x2 + xs'', xs'' >= 0, x >= 0, y >= 0, z' = 1, z1 = x, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, y) :|: z'' = 0, z2 = y, x >= 0, y >= 0, z' = 1, z1 = x
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> plusIter(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(x'', y'), 1 + x'', y, 1 + y') :|: z' = 1 + x'', z'' = y, y >= 0, z1 = 1 + y', y' >= 0, x'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, y, z) :|: z1 = z, z >= 0, z'' = y, y >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + x', y, 0) :|: z1 = 0, z' = 1 + x', z'' = y, x' >= 0, y >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   sum(z') -{ 1 }-> sumIter(xs, 0) :|: xs >= 0, z' = xs
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, x, plus(x, 1)) :|: x >= 0, z'' = x, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', x, plus(x, x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, x >= 0, z'' = x
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0


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

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

   { le }
   { tail }
   { isempty }
   { head }
   { a }
   { plusIter, ifPlus }
   { plus }
   { ifSum, sumIter }
   { sum }

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}

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

(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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: le
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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {le}, {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: ?, size: O(1) [2]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: le
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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(le(z' - 1, z1 - 1), 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {tail}, {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: 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: isempty
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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {isempty}, {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: ?, size: O(1) [2]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {head}, {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], 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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], 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: a
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 1

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {a}, {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  a: runtime: ?, size: O(1) [1]

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

(47) 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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

(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 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  a: runtime: O(1) [1], size: O(1) [1]

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  plusIter: runtime: ?, size: INF
  ifPlus: runtime: ?, size: INF

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

(53) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> z1 :|: z2 >= 0, z' = 2, z'' >= 0, z1 >= 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(z'', 1 + z1, 1 + z2) :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 1 }-> z1 :|: z'' >= 0, z' = 2, z1 >= 0, z2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(xs'', z2) :|: z'' = 1 + x2 + xs'', xs'' >= 0, z1 >= 0, z2 >= 0, z' = 1, x2 >= 0
   ifSum(z', z'', z1, z2) -{ 2 }-> sumIter(0, z2) :|: z'' = 0, z1 >= 0, z2 >= 0, z' = 1
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 2, z' - 1 >= 0, z'' - 1 >= 0
   le(z', z'') -{ 1 }-> 2 :|: z'' >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' - 1 >= 0
   le(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> plusIter(z', z'', 0) :|: z' >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 3 + z1 }-> ifPlus(s, 1 + (z' - 1), z'', 1 + (z1 - 1)) :|: s >= 0, s <= 2, z'' >= 0, z1 - 1 >= 0, z' - 1 >= 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(2, 0, z'', z1) :|: z1 >= 0, z'' >= 0, z' = 0
   plusIter(z', z'', z1) -{ 2 }-> ifPlus(1, 1 + (z' - 1), z'', 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(0, z', z'', z1) :|: z1 >= 0, z' >= 0, z'' >= 0
   sum(z') -{ 1 }-> sumIter(z', 0) :|: z' >= 0
   sumIter(z', z'') -{ 3 }-> ifSum(2, 0, z'', plus(z'', 1)) :|: z'' >= 0, z' = 0
   sumIter(z', z'') -{ 3 }-> ifSum(1, 1 + x1 + xs', z'', plus(z'', x1)) :|: x1 >= 0, z' = 1 + x1 + xs', xs' >= 0, z'' >= 0
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Function symbols to be analyzed: {plusIter,ifPlus}, {plus}, {ifSum,sumIter}, {sum}
Previous analysis results are:
  le: runtime: O(n^1) [2 + z''], size: O(1) [2]
  tail: runtime: O(1) [1], size: O(n^1) [z']
  isempty: runtime: O(1) [1], size: O(1) [2]
  head: runtime: O(1) [1], size: O(n^1) [1 + z']
  a: runtime: O(1) [1], size: O(1) [1]
  plusIter: runtime: INF, size: INF
  ifPlus: runtime: ?, size: INF

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

(55) 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:

   le(v0, v1) -> null_le [0]
   ifPlus(v0, v1, v2, v3) -> null_ifPlus [0]
   ifSum(v0, v1, v2, v3) -> null_ifSum [0]

And the following fresh constants:    null_le, null_ifPlus, null_ifSum

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

(56)
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:

   plus(x, y) -> plusIter(x, y, 0) [1]
   plusIter(x, y, z) -> ifPlus(le(x, z), x, y, z) [1]
   ifPlus(true, x, y, z) -> y [1]
   ifPlus(false, x, y, z) -> plusIter(x, s(y), s(z)) [1]
   le(s(x), 0) -> false [1]
   le(0, y) -> true [1]
   le(s(x), s(y)) -> le(x, y) [1]
   sum(xs) -> sumIter(xs, 0) [1]
   sumIter(xs, x) -> ifSum(isempty(xs), xs, x, plus(x, head(xs))) [1]
   ifSum(true, xs, x, y) -> x [1]
   ifSum(false, xs, x, y) -> sumIter(tail(xs), y) [1]
   isempty(nil) -> true [1]
   isempty(cons(x, xs)) -> false [1]
   head(nil) -> error [1]
   head(cons(x, xs)) -> x [1]
   tail(nil) -> nil [1]
   tail(cons(x, xs)) -> xs [1]
   a -> b [1]
   a -> c [1]
   le(v0, v1) -> null_le [0]
   ifPlus(v0, v1, v2, v3) -> null_ifPlus [0]
   ifSum(v0, v1, v2, v3) -> null_ifSum [0]

The TRS has the following type information:
plus :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
plusIter :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
0 :: 0:s:error:null_ifPlus:null_ifSum
ifPlus :: true:false:null_le -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
le :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> true:false:null_le
true :: true:false:null_le
false :: true:false:null_le
s :: 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
sum :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum
sumIter :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
ifSum :: true:false:null_le -> nil:cons -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum -> 0:s:error:null_ifPlus:null_ifSum
isempty :: nil:cons -> true:false:null_le
head :: nil:cons -> 0:s:error:null_ifPlus:null_ifSum
tail :: nil:cons -> nil:cons
nil :: nil:cons
cons :: 0:s:error:null_ifPlus:null_ifSum -> nil:cons -> nil:cons
error :: 0:s:error:null_ifPlus:null_ifSum
a :: b:c
b :: b:c
c :: b:c
null_le :: true:false:null_le
null_ifPlus :: 0:s:error:null_ifPlus:null_ifSum
null_ifSum :: 0:s:error:null_ifPlus:null_ifSum

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

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

   0 => 0
   true => 2
   false => 1
   nil => 0
   error => 1
   b => 0
   c => 1
   null_le => 0
   null_ifPlus => 0
   null_ifSum => 0

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

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

   a -{ 1 }-> 1 :|: 
   a -{ 1 }-> 0 :|: 
   head(z') -{ 1 }-> x :|: xs >= 0, z' = 1 + x + xs, x >= 0
   head(z') -{ 1 }-> 1 :|: z' = 0
   ifPlus(z', z'', z1, z2) -{ 1 }-> y :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   ifPlus(z', z'', z1, z2) -{ 1 }-> plusIter(x, 1 + y, 1 + z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   ifPlus(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   ifSum(z', z'', z1, z2) -{ 1 }-> x :|: xs >= 0, z2 = y, z' = 2, x >= 0, y >= 0, z'' = xs, z1 = x
   ifSum(z', z'', z1, z2) -{ 1 }-> sumIter(tail(xs), y) :|: xs >= 0, z2 = y, x >= 0, y >= 0, z'' = xs, z' = 1, z1 = x
   ifSum(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   isempty(z') -{ 1 }-> 2 :|: z' = 0
   isempty(z') -{ 1 }-> 1 :|: xs >= 0, z' = 1 + x + xs, x >= 0
   le(z', z'') -{ 1 }-> le(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   le(z', z'') -{ 1 }-> 2 :|: z'' = y, y >= 0, z' = 0
   le(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = 1 + x, x >= 0
   le(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> plusIter(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   plusIter(z', z'', z1) -{ 1 }-> ifPlus(le(x, z), x, y, z) :|: z1 = z, z >= 0, z' = x, z'' = y, x >= 0, y >= 0
   sum(z') -{ 1 }-> sumIter(xs, 0) :|: xs >= 0, z' = xs
   sumIter(z', z'') -{ 1 }-> ifSum(isempty(xs), xs, x, plus(x, head(xs))) :|: xs >= 0, x >= 0, z'' = x, z' = xs
   tail(z') -{ 1 }-> xs :|: xs >= 0, z' = 1 + x + xs, x >= 0
   tail(z') -{ 1 }-> 0 :|: z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(true, z0, z1, z2) -> c3
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), 0) -> c5
   LE(0, z0) -> c6
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUM(z0) -> c8(SUMITER(z0, 0))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(true, z0, z1, z2) -> c11
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
   ISEMPTY(nil) -> c13
   ISEMPTY(cons(z0, z1)) -> c14
   HEAD(nil) -> c15
   HEAD(cons(z0, z1)) -> c16
   TAIL(nil) -> c17
   TAIL(cons(z0, z1)) -> c18
   A -> c19
   A -> c20
K tuples:none
Defined Rule Symbols:   plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a

Defined Pair Symbols:   PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUM_1, SUMITER_2, IFSUM_4, ISEMPTY_1, HEAD_1, TAIL_1, A

Compound Symbols:   c1_1, c2_2, c3, c4_1, c5, c6, c7_1, c8_1, c9_2, c10_3, c11, c12_2, c13, c14, c15, c16, c17, c18, c19, c20


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

(61) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   SUM(z0) -> c8(SUMITER(z0, 0))
Removed 12 trailing nodes:
   IFSUM(true, z0, z1, z2) -> c11
   IFPLUS(true, z0, z1, z2) -> c3
   TAIL(nil) -> c17
   ISEMPTY(nil) -> c13
   LE(s(z0), 0) -> c5
   HEAD(cons(z0, z1)) -> c16
   LE(0, z0) -> c6
   A -> c20
   A -> c19
   TAIL(cons(z0, z1)) -> c18
   HEAD(nil) -> c15
   ISEMPTY(cons(z0, z1)) -> c14

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), ISEMPTY(z0))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)), HEAD(z0))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2), TAIL(z0))
K tuples:none
Defined Rule Symbols:   plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a

Defined Pair Symbols:   PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4

Compound Symbols:   c1_1, c2_2, c4_1, c7_1, c9_2, c10_3, c12_2


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

(63) CdtRhsSimplificationProcessorProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing tuple parts
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(z0, z1) -> plusIter(z0, z1, 0)
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
   a -> b
   a -> c
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
K tuples:none
Defined Rule Symbols:   plus_2, plusIter_3, ifPlus_4, le_2, sum_1, sumIter_2, ifSum_4, isempty_1, head_1, tail_1, a

Defined Pair Symbols:   PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4

Compound Symbols:   c1_1, c2_2, c4_1, c7_1, c9_1, c10_2, c12_1


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

(65) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   sum(z0) -> sumIter(z0, 0)
   sumIter(z0, z1) -> ifSum(isempty(z0), z0, z1, plus(z1, head(z0)))
   ifSum(true, z0, z1, z2) -> z1
   ifSum(false, z0, z1, z2) -> sumIter(tail(z0), z2)
   a -> b
   a -> c

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, PLUSITER_3, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4

Compound Symbols:   c1_1, c2_2, c4_1, c7_1, c9_1, c10_2, c12_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PLUSITER(z0, z1, z2) -> c2(IFPLUS(le(z0, z2), z0, z1, z2), LE(z0, z2)) by 
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0))
   PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0))
   PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0))
   PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3

Compound Symbols:   c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2


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

(69) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   PLUSITER(0, x1, z0) -> c2(IFPLUS(true, 0, x1, z0), LE(0, z0))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0), LE(s(z0), 0))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3

Compound Symbols:   c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3

Compound Symbols:   c1_1, c4_1, c7_1, c9_1, c10_2, c12_1, c2_2, c2_1


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(z0, z1) -> c9(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0)))) by 
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil))))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil))))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil))))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3

Compound Symbols:   c1_1, c4_1, c7_1, c10_2, c12_1, c2_2, c2_1, c9_1


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plus(x1, head(nil))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0)))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, SUMITER_2, IFSUM_4, PLUSITER_3

Compound Symbols:   c1_1, c4_1, c7_1, c10_2, c12_1, c2_2, c2_1, c9_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(z0, z1) -> c10(IFSUM(isempty(z0), z0, z1, plus(z1, head(z0))), PLUS(z1, head(z0))) by 
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plus(x1, head(nil))), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
K tuples:none
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2, c10_1


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   isempty(nil) -> true
   tail(nil) -> nil
   isempty(cons(z0, z1)) -> false
And the Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1 + x_2
   POL(LE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PLUSITER(x_1, x_2, x_3)) = x_3
   POL(SUMITER(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1 + x_2
   POL(error) = 0
   POL(false) = 0
   POL(head(x_1)) = 0
   POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_3 + x_4
   POL(isempty(x_1)) = 0
   POL(le(x_1, x_2)) = [1] + x_1 + x_2
   POL(nil) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = 0

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1)))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_2, c2_1, c9_1, c10_2, c10_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace PLUSITER(s(z0), x1, s(z1)) -> c2(IFPLUS(le(z0, z1), s(z0), x1, s(z1)), LE(s(z0), s(z1))) by 
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(0), x1, s(z0)) -> c2(IFPLUS(true, s(0), x1, s(z0)), LE(s(0), s(z0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0))) by 
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUMITER(nil, x1) -> c9(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(91) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plus(x1, error))) by 
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)))
   SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(93) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUMITER(nil, x0) -> c9(IFSUM(true, nil, x0, plus(x0, error)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0))) by 
   SUMITER(cons(z1, x1), z0) -> c9(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c2_2


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1))))) by 
   SUMITER(cons(x0, x1), z0) -> c9(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, plusIter(z0, head(x0), 0)), PLUS(z0, head(x0))) by 
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(IFSUM(true, nil, x1, plusIter(x1, head(nil), 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(103) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(105) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
We considered the (Usable) Rules:
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
And the Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_1
   POL(LE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PLUSITER(x_1, x_2, x_3)) = x_3
   POL(SUMITER(x_1, x_2)) = [1]
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_2
   POL(error) = 0
   POL(false) = [1]
   POL(head(x_1)) = 0
   POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_4
   POL(isempty(x_1)) = [1]
   POL(le(x_1, x_2)) = x_1
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1
   POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(s(x_1)) = 0
   POL(tail(x_1)) = [1] + x_1
   POL(true) = [1]

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(107) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plus(x1, error)), PLUS(x1, head(nil))) by 
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, head(nil)))
   SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(IFSUM(true, nil, x0, plus(x0, error)), PLUS(x0, head(nil)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, head(nil)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(111) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(nil, x0) -> c10(PLUS(x0, head(nil)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plus(x1, z0)), PLUS(x1, head(cons(z0, z1)))) by 
   SUMITER(cons(z1, x1), z0) -> c10(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0)), PLUS(z0, head(cons(z1, x1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_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.
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
We considered the (Usable) Rules:
   tail(cons(z0, z1)) -> z1
   tail(nil) -> nil
And the Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(IFPLUS(x_1, x_2, x_3, x_4)) = x_2
   POL(IFSUM(x_1, x_2, x_3, x_4)) = x_2
   POL(LE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(PLUSITER(x_1, x_2, x_3)) = x_3
   POL(SUMITER(x_1, x_2)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c12(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c7(x_1)) = x_1
   POL(c9(x_1)) = x_1
   POL(cons(x_1, x_2)) = [1] + x_2
   POL(error) = 0
   POL(false) = 0
   POL(head(x_1)) = 0
   POL(ifPlus(x_1, x_2, x_3, x_4)) = [1] + x_2 + x_3 + x_4
   POL(isempty(x_1)) = [1] + x_1
   POL(le(x_1, x_2)) = x_1
   POL(nil) = 0
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(plusIter(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(s(x_1)) = 0
   POL(tail(x_1)) = x_1
   POL(true) = [1]

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_2, c10_1, c2_2, c9_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plus(x1, head(cons(z0, z1)))), PLUS(x1, head(cons(z0, z1)))) by 
   SUMITER(cons(x0, x1), z0) -> c10(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0)), PLUS(z0, head(cons(x0, x1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) by 
   SUMITER(nil, x0) -> c10(PLUS(x0, error))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   IFPLUS(false, z0, z1, z2) -> c4(PLUSITER(z0, s(z1), s(z2)))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, IFPLUS_4, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2

Compound Symbols:   c1_1, c4_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2


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

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

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

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(0)) -> c2(IFPLUS(false, s(s(z0)), x1, s(0)), LE(s(s(z0)), s(0)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1


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

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

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(127) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
----------------------------------------

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(nil, x1) -> c9(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z1, x1), z0) -> c9(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x1) -> c9(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(137) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(x0, x1), z0) -> c9(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
S tuples:
   PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0))
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   PLUS_2, LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4

Compound Symbols:   c1_1, c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1


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

(139) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, z1) -> c1(PLUSITER(z0, z1, 0)) by 
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1))))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c2_2, c9_1, c10_2, c4_1, c_1, c1_1


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

(141) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(z0)), x1, s(s(z1))) -> c2(IFPLUS(le(z0, z1), s(s(z0)), x1, s(s(z1))), LE(s(s(z0)), s(s(z1)))) by 
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))

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

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))), LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2


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

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c9(IFSUM(false, cons(z0, z1), x2, plus(x2, z0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(147) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(x0), x1, s(x2)) -> c2(LE(s(x0), s(x2))) by 
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))

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

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil))) by SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(151) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(0), x1, s(z0)) -> c2(LE(s(0), s(z0))) by 
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(155) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFPLUS(false, s(s(x0)), x1, s(s(x2))) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(s(x2))))) by 
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(157) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(z0)), x1, s(0)) -> c(IFPLUS(false, s(s(z0)), x1, s(0))) by 
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(159) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IFPLUS(false, s(s(x0)), x1, s(0)) -> c4(PLUSITER(s(s(x0)), s(x1), s(s(0)))) by 
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, x1) -> c10(PLUS(x1, head(nil))) by SUMITER(nil, z0) -> c10(PLUS(z0, error))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c_1, c1_1, c2_2, c3_1


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

(165) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(z0)), x1, s(0)) -> c(LE(s(s(z0)), s(0))) by 
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))

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

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, x1) -> c10(IFSUM(isempty(nil), nil, x1, plusIter(x1, error, 0)), PLUS(x1, head(nil)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c1_1, c2_2, c3_1, c_1


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

(167) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, head(nil))) by SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
----------------------------------------

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c10_1, c9_1, c10_2, c4_1, c1_1, c2_2, c3_1, c_1


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

(169) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, x0) -> c10(PLUS(x0, head(nil))) by SUMITER(nil, z0) -> c10(PLUS(z0, error))
----------------------------------------

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), x1, plusIter(x1, z0, 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c2_2, c3_1, c_1


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

(171) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z1, x1), z0) -> c10(IFSUM(isempty(cons(z1, x1)), cons(z1, x1), z0, plusIter(z0, z1, 0)), PLUS(z0, head(cons(z1, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
----------------------------------------

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c2_2, c3_1, c_1


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

(173) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
----------------------------------------

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c2_2, c3_1, c_1


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

(175) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
----------------------------------------

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2)))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c2_2, c3_1, c_1


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

(177) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(IFPLUS(le(x0, s(x2)), s(s(x0)), s(x1), s(s(s(x2)))), LE(s(s(x0)), s(s(s(x2))))) by 
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0)))))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0)))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c2(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))), LE(s(s(x0)), s(s(s(0)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c3_1, c_1, c2_2


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

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c3_1, c_1, c2_2, c5_1


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

(181) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(x1), s(s(0)))) by 
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c3_1, c_1, c2_2, c5_1


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

(183) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c3(LE(s(s(x0)), s(s(0)))) by 
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))

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

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(cons(z0, z1), x1) -> c10(IFSUM(false, cons(z0, z1), x1, plusIter(x1, head(cons(z0, z1)), 0)), PLUS(x1, head(cons(z0, z1))))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(185) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(x0, x1), z0) -> c10(IFSUM(false, cons(x0, x1), z0, plusIter(z0, head(cons(x0, x1)), 0)), PLUS(z0, head(cons(x0, x1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
----------------------------------------

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1))))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(187) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), x2) -> c10(IFSUM(false, cons(z0, z1), x2, plus(x2, z0)), PLUS(x2, head(cons(z0, z1)))) by SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
----------------------------------------

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0))) by SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
K tuples:
   SUMITER(nil, x1) -> c10(PLUS(x1, head(nil)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(cons(x0, x1), x2) -> c10(PLUS(x2, head(cons(x0, x1))))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(191) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   SUMITER(nil, z0) -> c9(IFSUM(true, nil, z0, ifPlus(le(z0, 0), z0, error, 0)))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
K tuples:
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(193) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
----------------------------------------

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
K tuples:
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0))) by SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   le(s(z0), 0) -> false
   le(0, z0) -> true
   le(s(z0), s(z1)) -> le(z0, z1)
   isempty(nil) -> true
   isempty(cons(z0, z1)) -> false
   plus(z0, z1) -> plusIter(z0, z1, 0)
   head(nil) -> error
   head(cons(z0, z1)) -> z0
   plusIter(z0, z1, z2) -> ifPlus(le(z0, z2), z0, z1, z2)
   ifPlus(true, z0, z1, z2) -> z1
   ifPlus(false, z0, z1, z2) -> plusIter(z0, s(z1), s(z2))
   tail(nil) -> nil
   tail(cons(z0, z1)) -> z1
Tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
   SUMITER(nil, x0) -> c10(PLUS(x0, error))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, head(cons(z0, z1)), 0)))
   SUMITER(nil, z0) -> c9(IFSUM(isempty(nil), nil, z0, ifPlus(le(z0, 0), z0, error, 0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   SUMITER(cons(z0, z1), z2) -> c10(PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
S tuples:
   LE(s(z0), s(z1)) -> c7(LE(z0, z1))
   IFSUM(false, z0, z1, z2) -> c12(SUMITER(tail(z0), z2))
   PLUSITER(s(z0), x1, 0) -> c2(IFPLUS(false, s(z0), x1, 0))
   SUMITER(x0, z0) -> c9(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)))
   SUMITER(x0, z0) -> c10(IFSUM(isempty(x0), x0, z0, ifPlus(le(z0, 0), z0, head(x0), 0)), PLUS(z0, head(x0)))
   IFPLUS(false, s(x0), x1, 0) -> c4(PLUSITER(s(x0), s(x1), s(0)))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, plusIter(z2, z0, 0)))
   PLUS(s(y0), z1) -> c1(PLUSITER(s(y0), z1, 0))
   PLUSITER(s(x0), s(x1), s(0)) -> c2(LE(s(x0), s(0)))
   PLUSITER(s(s(x0)), s(x1), s(s(0))) -> c2(LE(s(s(x0)), s(s(0))))
   PLUSITER(s(s(x0)), s(x1), s(s(s(x2)))) -> c2(LE(s(s(x0)), s(s(s(x2)))))
   SUMITER(nil, z0) -> c10(IFSUM(isempty(nil), nil, z0, plusIter(z0, error, 0)), PLUS(z0, error))
   PLUSITER(s(0), s(x1), s(0)) -> c2(LE(s(0), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(isempty(cons(z0, z1)), cons(z0, z1), z2, plusIter(z2, z0, 0)), PLUS(z2, z0))
   IFPLUS(false, s(s(x0)), s(x1), s(s(s(x2)))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))))
   IFPLUS(false, s(s(x0)), s(x1), s(s(0))) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(IFPLUS(false, s(s(z0)), s(x1), s(0)))
   IFPLUS(false, s(s(x0)), s(x1), s(0)) -> c4(PLUSITER(s(s(x0)), s(s(x1)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plusIter(z2, head(cons(z0, z1)), 0)), PLUS(z2, z0))
   PLUSITER(s(s(z0)), s(x1), s(0)) -> c(LE(s(s(z0)), s(0)))
   SUMITER(cons(z0, z1), z2) -> c10(IFSUM(false, cons(z0, z1), z2, plus(z2, z0)), PLUS(z2, z0))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(s(x2))))) -> c2(IFPLUS(le(x0, s(s(x2))), s(s(x0)), s(s(x1)), s(s(s(s(x2))))), LE(s(s(x0)), s(s(s(s(x2))))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(IFPLUS(le(x0, s(0)), s(s(x0)), s(s(x1)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(s(0)))) -> c5(LE(s(s(x0)), s(s(s(0)))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(IFPLUS(le(x0, 0), s(s(x0)), s(s(x1)), s(s(0))))
   PLUSITER(s(s(x0)), s(s(x1)), s(s(0))) -> c3(LE(s(s(x0)), s(s(0))))
   SUMITER(cons(z0, z1), z2) -> c9(IFSUM(false, cons(z0, z1), z2, ifPlus(le(z2, 0), z2, z0, 0)))
K tuples:
   SUMITER(x0, x1) -> c10(PLUS(x1, head(x0)))
Defined Rule Symbols:   le_2, isempty_1, plus_2, head_1, plusIter_3, ifPlus_4, tail_1

Defined Pair Symbols:   LE_2, IFSUM_4, PLUSITER_3, SUMITER_2, IFPLUS_4, PLUS_2

Compound Symbols:   c7_1, c12_1, c2_1, c9_1, c10_2, c10_1, c4_1, c1_1, c_1, c2_2, c5_1, c3_1