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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 249 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 97 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 311 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 83 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 4977 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 1581 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (50) CdtProblem
    (51) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (56) CdtProblem
    (57) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (60) CdtProblem
    (61) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (64) CdtProblem
    (65) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem


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

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


The TRS R consists of the following rules:

   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fib(x) -> fibiter(x, 0, 0, s(0))
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
   if(false, b, c, x, y) -> x
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))

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

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

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


The TRS R consists of the following rules:

   plus(0', y) -> y
   plus(s(x), y) -> s(plus(x, y))
   lt(0', s(y)) -> true
   lt(x, 0') -> false
   lt(s(x), s(y)) -> lt(x, y)
   fib(x) -> fibiter(x, 0', 0', s(0'))
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
   if(false, b, c, x, y) -> x
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))

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

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

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


The TRS R consists of the following rules:

   plus(0, y) -> y
   plus(s(x), y) -> s(plus(x, y))
   lt(0, s(y)) -> true
   lt(x, 0) -> false
   lt(s(x), s(y)) -> lt(x, y)
   fib(x) -> fibiter(x, 0, 0, s(0))
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y)
   if(false, b, c, x, y) -> x
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y))

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

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

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


The TRS R consists of the following rules:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fib(x) -> fibiter(x, 0, 0, s(0)) [1]
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1]
   if(false, b, c, x, y) -> x [1]
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1]

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

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

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

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fib(x) -> fibiter(x, 0, 0, s(0)) [1]
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1]
   if(false, b, c, x, y) -> x [1]
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
fib :: 0:s -> 0:s
fibiter :: 0:s -> 0:s -> 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:

   fib_1
   fibiter_4
   if_5

(c) The following functions are completely defined:

   plus_2
   lt_2

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fib(x) -> fibiter(x, 0, 0, s(0)) [1]
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1]
   if(false, b, c, x, y) -> x [1]
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
fib :: 0:s -> 0:s
fibiter :: 0:s -> 0:s -> 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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fib(x) -> fibiter(x, 0, 0, s(0)) [1]
   fibiter(s(y'), 0, x, y) -> if(true, s(y'), 0, x, y) [2]
   fibiter(0, c, x, y) -> if(false, 0, c, x, y) [2]
   fibiter(s(y''), s(x'), x, y) -> if(lt(x', y''), s(y''), s(x'), x, y) [2]
   if(false, b, c, x, y) -> x [1]
   if(true, b, c, 0, y) -> fibiter(b, s(c), y, y) [2]
   if(true, b, c, s(x''), y) -> fibiter(b, s(c), y, s(plus(x'', y))) [2]

The TRS has the following type information:
plus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
fib :: 0:s -> 0:s
fibiter :: 0:s -> 0:s -> 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 }-> fibiter(x, 0, 0, 1 + 0) :|: x >= 0, z = x
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(x', y''), 1 + y'', 1 + x', x, y) :|: z = 1 + y'', z1 = y, z' = 1 + x', x' >= 0, x >= 0, y >= 0, z'' = x, y'' >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + y', 0, x, y) :|: z1 = y, x >= 0, y >= 0, z = 1 + y', z'' = x, y' >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, c, x, y) :|: z1 = y, c >= 0, x >= 0, y >= 0, z'' = x, z' = c, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> x :|: b >= 0, z2 = y, c >= 0, x >= 0, y >= 0, z' = b, z = 0, z'' = c, z1 = x
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(b, 1 + c, y, y) :|: b >= 0, z1 = 0, z2 = y, c >= 0, z = 1, y >= 0, z' = b, z'' = c
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(b, 1 + c, y, 1 + plus(x'', y)) :|: b >= 0, z2 = y, z1 = 1 + x'', c >= 0, z = 1, y >= 0, z' = b, x'' >= 0, z'' = c
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y


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

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

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

   fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0


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

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

   { lt }
   { plus }
   { if, fibiter }
   { fib }

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

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

   fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {fib}

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

(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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {fib}

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

(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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {lt}, {plus}, {if,fibiter}, {fib}
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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(lt(z' - 1, z - 1), 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {if,fibiter}, {fib}
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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {plus}, {if,fibiter}, {fib}
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: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

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

   fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, 1 + plus(z1 - 1, z2)) :|: z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0

Function symbols to be analyzed: {if,fibiter}, {fib}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z'], size: O(1) [1]
  plus: 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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

Computed SIZE bound using CoFloCo for: fibiter
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 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 1 }-> fibiter(z, 0, 0, 1 + 0) :|: z >= 0
   fibiter(z, z', z'', z1) -{ 3 + z }-> if(s', 1 + (z - 1), 1 + (z' - 1), z'', z1) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' >= 0, z1 >= 0, z - 1 >= 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(1, 1 + (z - 1), 0, z'', z1) :|: z'' >= 0, z1 >= 0, z - 1 >= 0, z' = 0
   fibiter(z, z', z'', z1) -{ 2 }-> if(0, 0, z', z'', z1) :|: z' >= 0, z'' >= 0, z1 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 1 }-> z1 :|: z' >= 0, z'' >= 0, z1 >= 0, z2 >= 0, z = 0
   if(z, z', z'', z1, z2) -{ 2 }-> fibiter(z', 1 + z'', z2, z2) :|: z' >= 0, z1 = 0, z'' >= 0, z = 1, z2 >= 0
   if(z, z', z'', z1, z2) -{ 2 + z1 }-> fibiter(z', 1 + z'', z2, 1 + s1) :|: s1 >= 0, s1 <= z1 - 1 + z2, z' >= 0, z'' >= 0, z = 1, z2 >= 0, z1 - 1 >= 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
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z - 1 >= 0, z' >= 0

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

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

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

And the following fresh constants: none

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

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

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

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   lt(0, s(y)) -> true [1]
   lt(x, 0) -> false [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   fib(x) -> fibiter(x, 0, 0, s(0)) [1]
   fibiter(b, c, x, y) -> if(lt(c, b), b, c, x, y) [1]
   if(false, b, c, x, y) -> x [1]
   if(true, b, c, x, y) -> fibiter(b, s(c), y, plus(x, y)) [1]

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

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

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

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

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

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

   fib(z) -{ 1 }-> fibiter(x, 0, 0, 1 + 0) :|: x >= 0, z = x
   fibiter(z, z', z'', z1) -{ 1 }-> if(lt(c, b), b, c, x, y) :|: b >= 0, z1 = y, c >= 0, x >= 0, y >= 0, z'' = x, z = b, z' = c
   if(z, z', z'', z1, z2) -{ 1 }-> x :|: b >= 0, z2 = y, c >= 0, x >= 0, y >= 0, z' = b, z = 0, z'' = c, z1 = x
   if(z, z', z'', z1, z2) -{ 1 }-> fibiter(b, 1 + c, y, plus(x, y)) :|: b >= 0, z2 = y, c >= 0, z = 1, x >= 0, y >= 0, z' = b, z'' = c, z1 = x
   lt(z, z') -{ 1 }-> lt(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   lt(z, z') -{ 1 }-> 1 :|: z' = 1 + y, y >= 0, z = 0
   lt(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fib(z0) -> fibiter(z0, 0, 0, s(0))
   fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
   if(false, z0, z1, z2, z3) -> z2
   if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))
Tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
S tuples:
   PLUS(0, z0) -> c
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(false, z0, z1, z2, z3) -> c7
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
K tuples:none
Defined Rule Symbols:   plus_2, lt_2, fib_1, fibiter_4, if_5

Defined Pair Symbols:   PLUS_2, LT_2, FIB_1, FIBITER_4, IF_5

Compound Symbols:   c, c1_1, c2, c3, c4_1, c5_1, c6_2, c7, c8_2


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(z0) -> c5(FIBITER(z0, 0, 0, s(0)))
Removed 4 trailing nodes:
   IF(false, z0, z1, z2, z3) -> c7
   LT(0, s(z0)) -> c2
   LT(z0, 0) -> c3
   PLUS(0, z0) -> c

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   fib(z0) -> fibiter(z0, 0, 0, s(0))
   fibiter(z0, z1, z2, z3) -> if(lt(z1, z0), z0, z1, z2, z3)
   if(false, z0, z1, z2, z3) -> z2
   if(true, z0, z1, z2, z3) -> fibiter(z0, s(z1), z3, plus(z2, z3))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
K tuples:none
Defined Rule Symbols:   plus_2, lt_2, fib_1, fibiter_4, if_5

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c8_2


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

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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c8_2


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBITER(z0, z1, z2, z3) -> c6(IF(lt(z1, z0), z0, z1, z2, z3), LT(z1, z0)) by 
   FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0)))
   FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0))
   FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0)))
   FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0))
   FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0)))
   FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0))
   FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_5, FIBITER_4

Compound Symbols:   c1_1, c4_1, c8_2, c6_2


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

(49) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIBITER(s(z0), 0, x2, x3) -> c6(IF(true, s(z0), 0, x2, x3), LT(0, s(z0)))
Removed 1 trailing nodes:
   FIBITER(0, z0, x2, x3) -> c6(IF(false, 0, z0, x2, x3), LT(z0, 0))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_5, FIBITER_4

Compound Symbols:   c1_1, c4_1, c8_2, c6_2


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

(51) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIBITER(s(z1), s(z0), x2, x3) -> c6(IF(lt(z0, z1), s(z1), s(z0), x2, x3), LT(s(z0), s(z1))) by 
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0)))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0)))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(0), s(z0), x2, x3) -> c6(IF(false, s(0), s(z0), x2, x3), LT(s(z0), s(0)))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_5, FIBITER_4

Compound Symbols:   c1_1, c4_1, c8_2, c6_2, c6_1


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
K tuples:none
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_5, FIBITER_4

Compound Symbols:   c1_1, c4_1, c8_2, c6_2, c6_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.
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
We considered the (Usable) Rules:none
And the Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(FIBITER(x_1, x_2, x_3, x_4)) = [1]
   POL(IF(x_1, x_2, x_3, x_4, x_5)) = [1]
   POL(LT(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c1(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1, x_2)) = x_1 + x_2
   POL(false) = [3]
   POL(lt(x_1, x_2)) = [3] + [3]x_1
   POL(plus(x_1, x_2)) = [2] + [3]x_2
   POL(s(x_1)) = 0
   POL(true) = [3]

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, IF_5, FIBITER_4

Compound Symbols:   c1_1, c4_1, c8_2, c6_2, c6_1


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

(57) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, z0, z1, z2, z3) -> c8(FIBITER(z0, s(z1), z3, plus(z2, z3)), PLUS(z2, z3)) by 
   IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c6_1, c8_2


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

(59) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   FIBITER(s(s(z0)), s(0), x2, x3) -> c6(IF(true, s(s(z0)), s(0), x2, x3), LT(s(0), s(s(z0))))
   FIBITER(s(0), s(z0), x2, x3) -> c6(LT(s(z0), s(0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(0), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)), PLUS(x1, x2))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c6_1, c8_2


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c6_1, c8_2, c_1


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

(63) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(PLUS(x1, x2))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c6_1, c8_2, c_1


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

(65) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
S tuples:
   PLUS(s(z0), z1) -> c1(PLUS(z0, z1))
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   PLUS_2, LT_2, FIBITER_4, IF_5

Compound Symbols:   c1_1, c4_1, c6_2, c6_1, c8_2, c_1


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

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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1))))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, FIBITER_4, IF_5, PLUS_2

Compound Symbols:   c4_1, c6_2, c6_1, c8_2, c_1, c1_1


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

(69) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FIBITER(s(s(z1)), s(s(z0)), x2, x3) -> c6(IF(lt(z0, z1), s(s(z1)), s(s(z0)), x2, x3), LT(s(s(z0)), s(s(z1)))) by 
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0), LT(s(s(0)), s(s(x0))))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, FIBITER_4, IF_5, PLUS_2

Compound Symbols:   c4_1, c6_1, c8_2, c_1, c1_1, c6_2


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

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, FIBITER_4, IF_5, PLUS_2

Compound Symbols:   c4_1, c6_1, c8_2, c_1, c1_1, c6_2, c2_1


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

(73) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
K tuples:
   FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0)))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, FIBITER_4, IF_5, PLUS_2

Compound Symbols:   c4_1, c6_1, c8_2, c_1, c1_1, c6_2, c2_1


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

(75) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FIBITER(s(x0), s(x1), x2, x3) -> c6(LT(s(x1), s(x0))) by 
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c8_2, c_1, c1_1, c6_2, c2_1, c6_1


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

(77) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace IF(true, s(s(x0)), s(s(x1)), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(x1))), x3, plus(x2, x3)), PLUS(x2, x3)) by 
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c8(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)), PLUS(x1, x2))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2, c3_1


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

(81) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c6_2, c2_1, c6_1, c8_2, c3_1


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

(83) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(IF(lt(s(x1), x0), s(s(x0)), s(s(s(x1))), x3, y0), LT(s(s(s(x1))), s(s(x0)))) by 
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0), LT(s(s(s(0))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2


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

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1


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

(87) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1


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

(89) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace FIBITER(s(s(x0)), s(s(s(x1))), x3, y0) -> c6(LT(s(s(s(x1))), s(s(x0)))) by 
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
S tuples:
   LT(s(z0), s(z1)) -> c4(LT(z0, z1))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   LT_2, IF_5, PLUS_2, FIBITER_4

Compound Symbols:   c4_1, c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LT(s(z0), s(z1)) -> c4(LT(z0, z1)) by 
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
S tuples:
   PLUS(s(s(y0)), z1) -> c1(PLUS(s(y0), z1))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, PLUS_2, FIBITER_4, LT_2

Compound Symbols:   c_1, c1_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1


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

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

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, LT_2, PLUS_2

Compound Symbols:   c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1, c1_1


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

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(PLUS(x1, x2)) by 
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1)))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, LT_2, PLUS_2

Compound Symbols:   c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c4_1, c1_1


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

(97) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace LT(s(s(y0)), s(s(y1))) -> c4(LT(s(y0), s(y1))) by 
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c6_1, c8_2, c3_1, c6_2, c5_1, c1_1, c4_1


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

(99) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c6(LT(s(s(0)), s(s(x0))))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(LT(s(s(0)), s(s(x0))))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(LT(s(s(s(0))), s(s(x0)))) by 
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1


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

(103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(LT(s(s(s(s(x1)))), s(s(x0)))) by 
   FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1)))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1)))))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0))))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c6_1, c1_1, c4_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c6(LT(s(s(s(0))), s(s(x0)))) by 
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1)))))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   lt(0, s(z0)) -> true
   lt(z0, 0) -> false
   lt(s(z0), s(z1)) -> lt(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> s(plus(z0, z1))
Tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1)))))
S tuples:
   IF(true, s(s(x0)), s(s(s(x1))), x2, x3) -> c8(FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, plus(x2, x3)), PLUS(x2, x3))
   FIBITER(s(s(x0)), s(s(s(s(x1)))), x3, y0) -> c6(IF(lt(s(s(x1)), x0), s(s(x0)), s(s(s(s(x1)))), x3, y0), LT(s(s(s(s(x1)))), s(s(x0))))
   PLUS(s(s(s(y0))), z1) -> c1(PLUS(s(s(y0)), z1))
   LT(s(s(s(y0))), s(s(s(y1)))) -> c4(LT(s(s(y0)), s(s(y1))))
K tuples:
   IF(true, s(s(x0)), s(0), x1, x2) -> c(FIBITER(s(s(x0)), s(s(0)), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(0)), x2, y0) -> c2(IF(lt(0, x0), s(s(x0)), s(s(0)), x2, y0))
   IF(true, s(s(x0)), s(s(0)), x1, x2) -> c3(FIBITER(s(s(x0)), s(s(s(0))), x2, plus(x1, x2)))
   FIBITER(s(s(x0)), s(s(s(0))), x2, y0) -> c5(IF(lt(s(0), x0), s(s(x0)), s(s(s(0))), x2, y0))
   IF(true, s(s(z0)), s(s(0)), s(s(s(y0))), z2) -> c3(PLUS(s(s(s(y0))), z2))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c5(LT(s(s(s(0))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(s(z1)))), z2, z3) -> c6(LT(s(s(s(s(z1)))), s(s(s(y1)))))
   FIBITER(s(s(s(y1))), s(s(s(0))), z1, z2) -> c6(LT(s(s(s(0))), s(s(s(y1)))))
Defined Rule Symbols:   lt_2, plus_2

Defined Pair Symbols:   IF_5, FIBITER_4, PLUS_2, LT_2

Compound Symbols:   c_1, c2_1, c8_2, c3_1, c6_2, c5_1, c1_1, c4_1, c6_1