KILLED
proof of input_uaq8MENqci.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 13 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 37 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), 344 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 266 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 107 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 30.1 s]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 4212 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 149 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 26 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 39 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 72 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 6 ms]
    (82) CdtProblem
    (83) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms]
    (86) CdtProblem
    (87) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 87 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) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (106) CdtProblem
    (107) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 63 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
    (114) CdtProblem
    (115) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (132) CdtProblem
    (133) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (136) CdtProblem
    (137) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtForwardInstantiationProof [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) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (170) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

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

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

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


The TRS R consists of the following rules:

   lt(0', s(x)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0') -> fib(0')
   fibo(s(0')) -> fib(s(0'))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0') -> s(0')
   fib(s(0')) -> s(0')
   fib(s(s(x))) -> if(true, 0', s(s(x)), 0', 0')
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0') -> x
   sum(x, s(y)) -> s(sum(x, y))

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

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

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


The TRS R consists of the following rules:

   lt(0, s(x)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0)
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c)
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b))
   sum(x, 0) -> x
   sum(x, s(y)) -> s(sum(x, y))

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

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

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


The TRS R consists of the following rules:

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fibo(0) -> fib(0) [1]
   fibo(s(0)) -> fib(s(0)) [1]
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1]
   fib(0) -> s(0) [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1]
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1]
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1]
   sum(x, 0) -> x [1]
   sum(x, s(y)) -> s(sum(x, y)) [1]

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

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

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fibo(0) -> fib(0) [1]
   fibo(s(0)) -> fib(s(0)) [1]
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1]
   fib(0) -> s(0) [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1]
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1]
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1]
   sum(x, 0) -> x [1]
   sum(x, s(y)) -> s(sum(x, y)) [1]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
fibo :: 0:s -> 0:s
fib :: 0:s -> 0:s
sum :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:
none

(c) The following functions are completely defined:

   fibo_1
   lt_2
   fib_1
   if_5
   sum_2

Due to the following rules being added:

   if(v0, v1, v2, v3, v4) -> 0 [0]

And the following fresh constants: none

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

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

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fibo(0) -> fib(0) [1]
   fibo(s(0)) -> fib(s(0)) [1]
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1]
   fib(0) -> s(0) [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1]
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1]
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1]
   sum(x, 0) -> x [1]
   sum(x, s(y)) -> s(sum(x, y)) [1]
   if(v0, v1, v2, v3, v4) -> 0 [0]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
fibo :: 0:s -> 0:s
fib :: 0:s -> 0:s
sum :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s

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

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

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

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fibo(0) -> fib(0) [1]
   fibo(s(0)) -> fib(s(0)) [1]
   fibo(s(s(0))) -> sum(fib(s(0)), fib(0)) [3]
   fibo(s(s(s(0)))) -> sum(sum(fibo(s(0)), fibo(0)), fib(s(0))) [3]
   fibo(s(s(s(s(x''))))) -> sum(sum(fibo(s(s(x''))), fibo(s(x''))), sum(fibo(s(x'')), fibo(x''))) [3]
   fib(0) -> s(0) [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1]
   if(true, c, s(s(x)), a, b) -> if(lt(c, s(x)), s(c), s(s(x)), b, c) [2]
   if(false, c, s(s(x)), 0, 0) -> sum(fib(0), fib(0)) [3]
   if(false, c, s(s(x)), 0, s(0)) -> sum(fib(0), fib(s(0))) [3]
   if(false, c, s(s(x)), 0, s(s(x2))) -> sum(fib(0), sum(fibo(s(x2)), fibo(x2))) [3]
   if(false, c, s(s(x)), s(0), 0) -> sum(fib(s(0)), fib(0)) [3]
   if(false, c, s(s(x)), s(0), s(0)) -> sum(fib(s(0)), fib(s(0))) [3]
   if(false, c, s(s(x)), s(0), s(s(x3))) -> sum(fib(s(0)), sum(fibo(s(x3)), fibo(x3))) [3]
   if(false, c, s(s(x)), s(s(x1)), 0) -> sum(sum(fibo(s(x1)), fibo(x1)), fib(0)) [3]
   if(false, c, s(s(x)), s(s(x1)), s(0)) -> sum(sum(fibo(s(x1)), fibo(x1)), fib(s(0))) [3]
   if(false, c, s(s(x)), s(s(x1)), s(s(x4))) -> sum(sum(fibo(s(x1)), fibo(x1)), sum(fibo(s(x4)), fibo(x4))) [3]
   sum(x, 0) -> x [1]
   sum(x, s(y)) -> s(sum(x, y)) [1]
   if(v0, v1, v2, v3, v4) -> 0 [0]

The TRS has the following type information:
lt :: 0:s -> 0:s -> true:false
0 :: 0:s
s :: 0:s -> 0:s
true :: true:false
false :: true:false
fibo :: 0:s -> 0:s
fib :: 0:s -> 0:s
sum :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s -> 0:s

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

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

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + x), 0, 0) :|: x >= 0, z = 1 + (1 + x)
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + x'')), fibo(1 + x'')), sum(fibo(1 + x''), fibo(x''))) :|: z = 1 + (1 + (1 + (1 + x''))), x'' >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), sum(fibo(1 + x4), fibo(x4))) :|: x1 >= 0, x4 >= 0, z2 = 1 + (1 + x4), c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1)
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), fib(0)) :|: x1 >= 0, c >= 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1)
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + x1), fibo(x1)), fib(1 + 0)) :|: z2 = 1 + 0, x1 >= 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z1 = 1 + (1 + x1)
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + x2), fibo(x2))) :|: z1 = 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, z2 = 1 + (1 + x2), x2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, c >= 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, c >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + x3), fibo(x3))) :|: z2 = 1 + (1 + x3), c >= 0, z1 = 1 + 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0, x3 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: c >= 0, z1 = 1 + 0, z2 = 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, c >= 0, z1 = 1 + 0, x >= 0, z'' = 1 + (1 + x), z' = c, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(c, 1 + x), 1 + c, 1 + (1 + x), b, c) :|: b >= 0, z2 = b, c >= 0, a >= 0, z = 1, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   sum(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x


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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0


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

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

   { lt }
   { sum }
   { fib, if, fibo }

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo}

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

(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:

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

Function symbols to be analyzed: {lt}, {sum}, {fib,if,fibo}
Previous analysis results are:
  lt: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> if(lt(z', 1 + (z'' - 2)), 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 1 }-> lt(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

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

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> 1 + sum(z, z' - 1) :|: z >= 0, z' - 1 >= 0

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

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: fib
after applying outer abstraction to obtain an ITS,
resulting in: EXP with polynomial bound: ?

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

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> if(1, 0, 1 + (1 + (z - 2)), 0, 0) :|: z - 2 >= 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + 0), fibo(0)), fib(1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fibo(z) -{ 3 }-> sum(sum(fibo(1 + (1 + (z - 4))), fibo(1 + (z - 4))), sum(fibo(1 + (z - 4)), fibo(z - 4))) :|: z - 4 >= 0
   fibo(z) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z = 1 + (1 + 0)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 - 2 >= 0, z2 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(0)) :|: z1 - 2 >= 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(sum(fibo(1 + (z1 - 2)), fibo(z1 - 2)), fib(1 + 0)) :|: z2 = 1 + 0, z1 - 2 >= 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z1 = 0, z' >= 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(0)) :|: z1 = 0, z' >= 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(0), fib(1 + 0)) :|: z1 = 0, z2 = 1 + 0, z' >= 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), sum(fibo(1 + (z2 - 2)), fibo(z2 - 2))) :|: z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0, z2 - 2 >= 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(0)) :|: z' >= 0, z1 = 1 + 0, z2 = 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 }-> sum(fib(1 + 0), fib(1 + 0)) :|: z2 = 1 + 0, z' >= 0, z1 = 1 + 0, z'' - 2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 3 + z'' }-> if(s', 1 + z', 1 + (1 + (z'' - 2)), z2, z') :|: s' >= 0, s' <= 1, z2 >= 0, z' >= 0, z1 >= 0, z = 1, z'' - 2 >= 0
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z >= 0, z2 >= 0, z' >= 0, z'' >= 0, z1 >= 0
   lt(z, z') -{ 2 + z' }-> s :|: s >= 0, s <= 1, z - 1 >= 0, z' - 1 >= 0
   lt(z, z') -{ 1 }-> 1 :|: z' - 1 >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   sum(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   sum(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0

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

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

(37) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   if(v0, v1, v2, v3, v4) -> null_if [0]
   lt(v0, v1) -> null_lt [0]
   fibo(v0) -> null_fibo [0]
   fib(v0) -> null_fib [0]
   sum(v0, v1) -> null_sum [0]

And the following fresh constants:    null_if, null_lt, null_fibo, null_fib, null_sum

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

(38)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   lt(0, s(x)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fibo(0) -> fib(0) [1]
   fibo(s(0)) -> fib(s(0)) [1]
   fibo(s(s(x))) -> sum(fibo(s(x)), fibo(x)) [1]
   fib(0) -> s(0) [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(x))) -> if(true, 0, s(s(x)), 0, 0) [1]
   if(true, c, s(s(x)), a, b) -> if(lt(s(c), s(s(x))), s(c), s(s(x)), b, c) [1]
   if(false, c, s(s(x)), a, b) -> sum(fibo(a), fibo(b)) [1]
   sum(x, 0) -> x [1]
   sum(x, s(y)) -> s(sum(x, y)) [1]
   if(v0, v1, v2, v3, v4) -> null_if [0]
   lt(v0, v1) -> null_lt [0]
   fibo(v0) -> null_fibo [0]
   fib(v0) -> null_fib [0]
   sum(v0, v1) -> null_sum [0]

The TRS has the following type information:
lt :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> true:false:null_lt
0 :: 0:s:null_if:null_fibo:null_fib:null_sum
s :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum
true :: true:false:null_lt
false :: true:false:null_lt
fibo :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum
fib :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum
sum :: 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum
if :: true:false:null_lt -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum -> 0:s:null_if:null_fibo:null_fib:null_sum
null_if :: 0:s:null_if:null_fibo:null_fib:null_sum
null_lt :: true:false:null_lt
null_fibo :: 0:s:null_if:null_fibo:null_fib:null_sum
null_fib :: 0:s:null_if:null_fibo:null_fib:null_sum
null_sum :: 0:s:null_if:null_fibo:null_fib:null_sum

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

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

   0 => 0
   true => 2
   false => 1
   null_if => 0
   null_lt => 0
   null_fibo => 0
   null_fib => 0
   null_sum => 0

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

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

   fib(z) -{ 1 }-> if(2, 0, 1 + (1 + x), 0, 0) :|: x >= 0, z = 1 + (1 + x)
   fib(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fibo(z) -{ 1 }-> sum(fibo(1 + x), fibo(x)) :|: x >= 0, z = 1 + (1 + x)
   fibo(z) -{ 1 }-> fib(0) :|: z = 0
   fibo(z) -{ 1 }-> fib(1 + 0) :|: z = 1 + 0
   fibo(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   if(z, z', z'', z1, z2) -{ 1 }-> sum(fibo(a), fibo(b)) :|: b >= 0, z2 = b, c >= 0, a >= 0, z = 1, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a
   if(z, z', z'', z1, z2) -{ 1 }-> if(lt(1 + c, 1 + (1 + x)), 1 + c, 1 + (1 + x), b, c) :|: z = 2, b >= 0, z2 = b, c >= 0, a >= 0, x >= 0, z'' = 1 + (1 + x), z' = c, z1 = a
   if(z, z', z'', z1, z2) -{ 0 }-> 0 :|: z1 = v3, v0 >= 0, v4 >= 0, z'' = v2, v1 >= 0, z = v0, z' = v1, z2 = v4, v2 >= 0, v3 >= 0
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 2 :|: z' = 1 + x, x >= 0, z = 0
   lt(z, z') -{ 1 }-> 1 :|: x >= 0, z = x, z' = 0
   lt(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   sum(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   sum(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   sum(z, z') -{ 1 }-> 1 + sum(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
Tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
S tuples:
   LT(0, s(z0)) -> c
   LT(z0, 0) -> c1
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(0) -> c3(FIB(0))
   FIBO(s(0)) -> c4(FIB(s(0)))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   FIB(0) -> c7
   FIB(s(0)) -> c8
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, 0) -> c13
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
K tuples:none
Defined Rule Symbols:   lt_2, fibo_1, fib_1, if_5, sum_2

Defined Pair Symbols:   LT_2, FIBO_1, FIB_1, IF_5, SUM_2

Compound Symbols:   c, c1, c2_1, c3_1, c4_1, c5_2, c6_2, c7, c8, c9_1, c10_2, c11_2, c12_2, c13, c14_1


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(s(s(z0))) -> c9(IF(true, 0, s(s(z0)), 0, 0))
Removed 7 trailing nodes:
   FIB(s(0)) -> c8
   FIB(0) -> c7
   LT(z0, 0) -> c1
   FIBO(0) -> c3(FIB(0))
   LT(0, s(z0)) -> c
   FIBO(s(0)) -> c4(FIB(s(0)))
   SUM(z0, 0) -> c13

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   IF(false, z0, s(s(z1)), z2, z3) -> c11(SUM(fibo(z2), fibo(z3)), FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c12(SUM(fibo(z2), fibo(z3)), FIBO(z3))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
K tuples:none
Defined Rule Symbols:   lt_2, fibo_1, fib_1, if_5, sum_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c5_2, c6_2, c10_2, c11_2, c12_2, c14_1


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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fibo(0) -> fib(0)
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fib(0) -> s(0)
   fib(s(0)) -> s(0)
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
K tuples:none
Defined Rule Symbols:   lt_2, fibo_1, fib_1, if_5, sum_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c5_2, c6_2, c10_2, c14_1, c_1


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

(47) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fib(s(s(z0))) -> if(true, 0, s(s(z0)), 0, 0)
   if(true, z0, s(s(z1)), z2, z3) -> if(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0)
   if(false, z0, s(s(z1)), z2, z3) -> sum(fibo(z2), fibo(z3))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
K tuples:none
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c5_2, c6_2, c10_2, c14_1, c_1


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

(49) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [3]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [3] + x_1 + [3]x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(fib(x_1)) = [3]
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(sum(x_1, x_2)) = [3]
   POL(true) = [1]

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0)))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c5_2, c6_2, c10_2, c14_1, c_1


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(z0))) -> c5(SUM(fibo(s(z0)), fibo(z0)), FIBO(s(z0))) by 
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0)))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0)))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)), FIBO(s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)), FIBO(s(0)))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c6_2, c10_2, c14_1, c_1, c5_2


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c6_2, c10_2, c14_1, c_1, c5_2, c5_1


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

(55) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(sum(x_1, x_2)) = 0
   POL(true) = [1]

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, FIBO_1, IF_5, SUM_2

Compound Symbols:   c2_1, c6_2, c10_2, c14_1, c_1, c5_2, c5_1


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

(57) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(z0))) -> c6(SUM(fibo(s(z0)), fibo(z0)), FIBO(z0)) by 
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)), FIBO(0))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)), FIBO(0))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, IF_5, SUM_2, FIBO_1

Compound Symbols:   c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2


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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c5(SUM(fibo(s(s(0))), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, IF_5, SUM_2, FIBO_1

Compound Symbols:   c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, IF_5, SUM_2, FIBO_1

Compound Symbols:   c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_2 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = [1]
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = [1]
   POL(s(x_1)) = 0
   POL(sum(x_1, x_2)) = [1]
   POL(true) = [1]

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1))))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, IF_5, SUM_2, FIBO_1

Compound Symbols:   c2_1, c10_2, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, z0, s(s(z1)), z2, z3) -> c10(IF(lt(s(z0), s(s(z1))), s(z0), s(s(z1)), z3, z0), LT(s(z0), s(s(z1)))) by 
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3))) by 
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) by 
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_2, c5_1, c6_2, c6_1, c1_1, c10_2


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(z0)))) -> c5(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(s(z0)))) by 
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c5(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1


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

(75) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = 0
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = [1]
   POL(s(x_1)) = [1]
   POL(sum(x_1, x_2)) = 0
   POL(true) = [1]

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0))) by 
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c5_1, c6_2, c6_1, c1_1, c10_2, c5_2, c3_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0))) by 
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(s(z0))))) -> c6(SUM(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) by 
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1


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

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c4_1


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

(85) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_1 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(false) = 0
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = 0
   POL(s(x_1)) = [1]
   POL(sum(x_1, x_2)) = 0
   POL(true) = 0

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0)))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_2, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c4_1


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

(87) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(z0)))) -> c6(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), FIBO(s(z0))) by 
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))), FIBO(s(0)))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1


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

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(0))))) -> c5(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(0)))))) -> c6(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c6(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))), FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1


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

(93) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
We considered the (Usable) Rules:none
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1] + x_2 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c10(x_1, x_2)) = x_1 + x_2
   POL(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(false) = [1]
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = 0
   POL(sum(x_1, x_2)) = 0
   POL(true) = [1]

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1


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

(95) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0)))) by 
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0))) by 
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c6_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c7_1


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

(99) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0))) by 
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1


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

(101) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0)))) by 
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c10_2, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(true, z0, s(s(x1)), x2, x3) -> c10(IF(lt(z0, s(x1)), s(z0), s(s(x1)), x3, z0), LT(s(z0), s(s(x1)))) by 
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(SUM(fibo(z2), fibo(z3)))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(105) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF(true, 0, s(s(z0)), x2, x3) -> c10(IF(true, s(0), s(s(z0)), x3, 0), LT(s(0), s(s(z0))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(107) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
We considered the (Usable) Rules:
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(z0, 0) -> false
And the Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBO(x_1)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = x_1 + x_2 + x_3
   POL(LT(x_1, x_2)) = 0
   POL(SUM(x_1, x_2)) = 0
   POL(c(x_1)) = 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(c14(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(c5(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(false) = 0
   POL(fib(x_1)) = 0
   POL(fibo(x_1)) = 0
   POL(lt(x_1, x_2)) = 0
   POL(s(x_1)) = [1]
   POL(sum(x_1, x_2)) = 0
   POL(true) = 0

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), fib(s(0)))) by 
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(111) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), x2, s(s(z0))) -> c(SUM(fibo(x2), sum(fibo(s(z0)), fibo(z0)))) by 
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(113) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), fib(0))) by 
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(115) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), s(0), x3) -> c(SUM(fib(s(0)), fibo(x3))) by 
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(0), s(s(z0))) -> c(SUM(fib(s(0)), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(117) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), s(s(z0)), x3) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fibo(x3))) by 
   IF(false, x0, s(s(x1)), s(s(x2)), s(0)) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(x2)), s(s(z0))) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), s(s(x2)), 0) -> c(SUM(sum(fibo(s(x2)), fibo(x2)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))

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

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace IF(false, x0, s(s(x1)), 0, x3) -> c(SUM(fib(0), fibo(x3))) by 
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(s(z0))) -> c(SUM(fib(0), sum(fibo(s(z0)), fibo(z0))))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(121) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0))))))) by FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
----------------------------------------

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))), FIBO(s(s(s(s(0)))))) by FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(127) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z2)) by 
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, IF_5, FIBO_1

Compound Symbols:   c2_1, c14_1, c_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1


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

(129) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(false, z0, s(s(z1)), z2, z3) -> c(FIBO(z3)) by 
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))

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

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_2, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1


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

(133) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
----------------------------------------

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c5_2, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c5_2, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1


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

(137) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
----------------------------------------

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(139) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
----------------------------------------

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(143) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0)))) by FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
----------------------------------------

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(145) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fibo(s(0)), s(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
----------------------------------------

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(fib(s(0)), fib(0))) by FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(151) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(0))) -> c5(SUM(s(0), fibo(0))) by FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
----------------------------------------

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(153) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0)))))) by FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
----------------------------------------

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(155) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(fibo(s(s(s(s(z0))))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0))))) by FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
----------------------------------------

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(157) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0)))) by FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
----------------------------------------

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
S tuples:
   LT(s(z0), s(z1)) -> c2(LT(z0, z1))
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   LT_2, SUM_2, FIBO_1, IF_5

Compound Symbols:   c2_1, c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2


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

(159) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LT(s(z0), s(z1)) -> c2(LT(z0, z1)) by 
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))

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

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
S tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   SUM_2, FIBO_1, IF_5, LT_2

Compound Symbols:   c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0)))) by FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
S tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   SUM_2, FIBO_1, IF_5, LT_2

Compound Symbols:   c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1


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

(163) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0)))) by FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0))))
----------------------------------------

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c6(SUM(sum(fibo(s(s(x0))), fibo(s(x0))), sum(fibo(s(x0)), fibo(x0))), FIBO(s(s(x0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0))))
S tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   SUM_2, FIBO_1, IF_5, LT_2

Compound Symbols:   c14_1, c1_1, c5_1, c3_1, c6_2, c4_1, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0)))) by FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0))))
S tuples:
   SUM(z0, s(z1)) -> c14(SUM(z0, z1))
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   SUM_2, FIBO_1, IF_5, LT_2

Compound Symbols:   c14_1, c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SUM(z0, s(z1)) -> c14(SUM(z0, z1)) by 
   SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1)))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0))))
   SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1)))
S tuples:
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   FIBO_1, IF_5, LT_2, SUM_2

Compound Symbols:   c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1, c14_1


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

(169) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 7 trailing nodes:
   IF(false, x0, s(s(x1)), x2, 0) -> c(SUM(fibo(x2), s(0)))
   IF(false, x0, s(s(x1)), x2, s(0)) -> c(SUM(fibo(x2), s(0)))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fibo(0)), s(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c6(SUM(fibo(s(0)), s(0)))
   FIBO(s(s(s(0)))) -> c6(SUM(fibo(s(s(0))), s(0)))
   FIBO(s(s(0))) -> c5(SUM(fib(s(0)), s(0)))

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fibo(s(0)) -> fib(s(0))
   fibo(s(s(z0))) -> sum(fibo(s(z0)), fibo(z0))
   fibo(0) -> fib(0)
   fib(s(0)) -> s(0)
   fib(0) -> s(0)
   sum(z0, 0) -> z0
   sum(z0, s(z1)) -> s(sum(z0, z1))
   lt(s(z0), s(z1)) -> lt(z0, z1)
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
Tuples:
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   FIBO(s(s(0))) -> c6(SUM(fib(s(0)), fib(0)))
   FIBO(s(s(0))) -> c6(SUM(s(0), fibo(0)))
   FIBO(s(s(s(0)))) -> c1(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, x0, s(s(x1)), s(0), s(0)) -> c(SUM(fib(s(0)), fib(s(0))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(0)) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(s(0))))
   IF(false, x0, s(s(x1)), 0, s(0)) -> c(SUM(fib(0), fib(s(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(0)))) -> c(SUM(fibo(x2), sum(fibo(s(s(0))), fib(s(0)))))
   IF(false, x0, s(s(x1)), x2, s(s(s(s(z0))))) -> c(SUM(fibo(x2), sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0)))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fibo(s(0)), fib(0))))
   IF(false, x0, s(s(x1)), x2, s(s(0))) -> c(SUM(fibo(x2), sum(fib(s(0)), fibo(0))))
   IF(false, x0, s(s(x1)), x2, s(s(s(z0)))) -> c(SUM(fibo(x2), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))))
   IF(false, x0, s(s(x1)), s(0), s(s(x3))) -> c(SUM(fib(s(0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(s(z0)), s(s(x3))) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), 0, s(s(x3))) -> c(SUM(fib(0), sum(fibo(s(x3)), fibo(x3))))
   IF(false, x0, s(s(x1)), s(0), 0) -> c(SUM(fib(s(0)), fib(0)))
   IF(false, x0, s(s(x1)), s(s(z0)), 0) -> c(SUM(sum(fibo(s(z0)), fibo(z0)), fib(0)))
   IF(false, x0, s(s(x1)), 0, 0) -> c(SUM(fib(0), fib(0)))
   IF(false, x0, s(s(x1)), s(0), x2) -> c(SUM(s(0), fibo(x2)))
   IF(false, x0, s(s(x1)), s(s(s(0))), x3) -> c(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(s(z0)))), x3) -> c(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fibo(s(0)), fib(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(0)), x3) -> c(SUM(sum(fib(s(0)), fibo(0)), fibo(x3)))
   IF(false, x0, s(s(x1)), s(s(s(z0))), x3) -> c(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(x3)))
   IF(false, x0, s(s(x1)), 0, x2) -> c(SUM(s(0), fibo(x2)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fibo(s(0)), fib(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c3(SUM(sum(fib(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(0))) -> c5(SUM(s(0), fib(0)))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), s(0))))
   FIBO(s(s(s(s(0))))) -> c4(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fib(0))))
   SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1)))
S tuples:
   FIBO(s(s(s(0)))) -> c1(FIBO(s(s(0))))
   FIBO(s(s(s(s(x0))))) -> c5(FIBO(s(s(s(x0)))))
   FIBO(s(s(s(0)))) -> c3(FIBO(s(s(0))))
   FIBO(s(s(s(s(0))))) -> c4(FIBO(s(s(0))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), fibo(s(s(s(z0))))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), fibo(s(s(z0)))), FIBO(s(s(z0))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(FIBO(s(s(s(0)))))
   FIBO(s(s(s(s(0))))) -> c7(FIBO(s(s(0))))
   IF(true, s(z0), s(s(z1)), x2, x3) -> c10(IF(lt(z0, z1), s(s(z0)), s(s(z1)), x3, s(z0)), LT(s(s(z0)), s(s(z1))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c5(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(s(z0)))))))
   FIBO(s(s(s(s(s(0)))))) -> c5(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), s(0))), FIBO(s(s(s(s(0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(z0))))) -> c5(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(s(z0)))))) -> c5(SUM(sum(fibo(s(s(s(z0)))), sum(fibo(s(z0)), fibo(z0))), sum(fibo(s(s(z0))), fibo(s(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(s(z0))))))) -> c6(SUM(fibo(s(s(s(s(s(z0)))))), sum(sum(fibo(s(s(z0))), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0)))), FIBO(s(s(s(s(z0))))))
   FIBO(s(s(s(s(s(z0)))))) -> c6(SUM(sum(fibo(s(s(s(z0)))), fibo(s(s(z0)))), sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0)))), FIBO(s(s(s(z0)))))
   FIBO(s(s(s(s(z0))))) -> c6(SUM(sum(sum(fibo(s(z0)), fibo(z0)), fibo(s(z0))), sum(fibo(s(z0)), fibo(z0))), FIBO(s(s(z0))))
   LT(s(s(y0)), s(s(y1))) -> c2(LT(s(y0), s(y1)))
   SUM(z0, s(s(y1))) -> c14(SUM(z0, s(y1)))
K tuples:
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fibo(0)), fib(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fibo(s(0)), fib(0)), fibo(s(0))))
   FIBO(s(s(s(0)))) -> c6(SUM(sum(fib(s(0)), fibo(0)), fibo(s(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fibo(s(0)), fib(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(fibo(s(s(s(0)))), sum(fib(s(0)), fibo(0))))
   FIBO(s(s(s(s(0))))) -> c7(SUM(sum(fibo(s(s(0))), fib(s(0))), fibo(s(s(0)))))
   FIBO(s(s(s(s(s(0)))))) -> c7(SUM(fibo(s(s(s(s(0))))), sum(fibo(s(s(0))), fib(s(0)))))
   IF(true, x0, s(s(x1)), x2, x3) -> c10(LT(s(x0), s(s(x1))))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(x3))
   IF(false, s(s(x0)), s(s(x1)), x3, s(x0)) -> c(FIBO(s(x0)))
Defined Rule Symbols:   fibo_1, fib_1, sum_2, lt_2

Defined Pair Symbols:   FIBO_1, IF_5, LT_2, SUM_2

Compound Symbols:   c1_1, c5_1, c3_1, c4_1, c6_2, c6_1, c7_1, c10_2, c10_1, c_1, c5_2, c2_1, c14_1