KILLED
proof of input_cWAIvbvHkW.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxWeightedTrs
    (9) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedTrs
    (11) CompletionProof [UPPER BOUND(ID), 0 ms]
    (12) CpxTypedWeightedCompleteTrs
        (13) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (14) CpxTypedWeightedCompleteTrs
        (15) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (16) CpxRNTS
        (17) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (20) CpxRNTS
        (21) ResultPropagationProof [UPPER BOUND(ID), 3 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 228 ms]
        (24) CpxRNTS
        (25) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
        (26) CpxRNTS
        (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 133 ms]
        (30) CpxRNTS
        (31) IntTrsBoundProof [UPPER BOUND(ID), 46 ms]
        (32) CpxRNTS
        (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 91 ms]
        (36) CpxRNTS
        (37) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
        (38) CpxRNTS
        (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 1458 ms]
        (42) CpxRNTS
        (43) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
        (44) CpxRNTS
    (45) CompletionProof [UPPER BOUND(ID), 0 ms]
    (46) CpxTypedWeightedCompleteTrs
        (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
(49) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(50) CdtProblem
    (51) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (56) CdtProblem
    (57) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 106 ms]
    (62) CdtProblem
    (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (104) CdtProblem
    (105) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) 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:

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(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:

   fib(0') -> 0'
   fib(s(0')) -> s(0')
   fib(s(s(0'))) -> s(0')
   fib(s(s(x))) -> sp(g(x))
   g(0') -> pair(s(0'), 0')
   g(s(0')) -> pair(s(0'), s(0'))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +'(x, y)
   np(pair(x, y)) -> pair(+'(x, y), x)
   +'(x, 0') -> x
   +'(x, s(y)) -> s(+'(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:

   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(x))) -> sp(g(x))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(x)) -> np(g(x))
   sp(pair(x, y)) -> +(x, y)
   np(pair(x, y)) -> pair(+(x, y), x)
   +(x, 0) -> x
   +(x, s(y)) -> s(+(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:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(x))) -> sp(g(x)) [1]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(x)) -> np(g(x)) [1]
   sp(pair(x, y)) -> +(x, y) [1]
   np(pair(x, y)) -> pair(+(x, y), x) [1]
   +(x, 0) -> x [1]
   +(x, s(y)) -> s(+(x, y)) [1]

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

(7) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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

(8)
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:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(x))) -> sp(g(x)) [1]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(x)) -> np(g(x)) [1]
   sp(pair(x, y)) -> plus(x, y) [1]
   np(pair(x, y)) -> pair(plus(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

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

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

(10)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(x))) -> sp(g(x)) [1]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(x)) -> np(g(x)) [1]
   sp(pair(x, y)) -> plus(x, y) [1]
   np(pair(x, y)) -> pair(plus(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]

The TRS has the following type information:
fib :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sp :: pair -> 0:s
g :: 0:s -> pair
pair :: 0:s -> 0:s -> pair
np :: pair -> pair
plus :: 0:s -> 0:s -> 0:s

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

(11) 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
   sp_1

(c) The following functions are completely defined:

   g_1
   np_1
   plus_2

Due to the following rules being added:

   np(v0) -> const [0]

And the following fresh constants:    const

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

(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:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(x))) -> sp(g(x)) [1]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(x)) -> np(g(x)) [1]
   sp(pair(x, y)) -> plus(x, y) [1]
   np(pair(x, y)) -> pair(plus(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   np(v0) -> const [0]

The TRS has the following type information:
fib :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sp :: pair:const -> 0:s
g :: 0:s -> pair:const
pair :: 0:s -> 0:s -> pair:const
np :: pair:const -> pair:const
plus :: 0:s -> 0:s -> 0:s
const :: pair:const

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

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

(14)
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:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(0))) -> sp(pair(s(0), 0)) [2]
   fib(s(s(s(0)))) -> sp(pair(s(0), s(0))) [2]
   fib(s(s(s(x')))) -> sp(np(g(x'))) [2]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(0)) -> np(pair(s(0), 0)) [2]
   g(s(s(0))) -> np(pair(s(0), s(0))) [2]
   g(s(s(x''))) -> np(np(g(x''))) [2]
   sp(pair(x, y)) -> plus(x, y) [1]
   np(pair(x, y)) -> pair(plus(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   np(v0) -> const [0]

The TRS has the following type information:
fib :: 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
sp :: pair:const -> 0:s
g :: 0:s -> pair:const
pair :: 0:s -> 0:s -> pair:const
np :: pair:const -> pair:const
plus :: 0:s -> 0:s -> 0:s
const :: pair:const

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

(15) 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
   const => 0

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

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

   fib(z) -{ 2 }-> sp(np(g(x'))) :|: z = 1 + (1 + (1 + x')), x' >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(x''))) :|: x'' >= 0, z = 1 + (1 + x'')
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0


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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0


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

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

   { plus }
   { np }
   { sp }
   { g }
   { fib }

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

(20)
Obligation:
Complexity RNTS consisting of the following rules:

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {plus}, {np}, {sp}, {g}, {fib}

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {plus}, {np}, {sp}, {g}, {fib}

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

(23) 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'

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0

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

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

(25) 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'

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 }-> 1 + plus(z, z' - 1) :|: z >= 0, z' - 1 >= 0
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0

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

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: np
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 2 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   fib(z) -{ 2 }-> sp(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

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

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

   fib(z) -{ 5 }-> s3 :|: s3 >= 0, s3 <= 1 + (1 + 0) + 0, z = 1 + (1 + 0)
   fib(z) -{ 6 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + 0) + (1 + 0), z = 1 + (1 + (1 + 0))
   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

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

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 5 }-> s3 :|: s3 >= 0, s3 <= 1 + (1 + 0) + 0, z = 1 + (1 + 0)
   fib(z) -{ 6 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + 0) + (1 + 0), z = 1 + (1 + (1 + 0))
   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {g}, {fib}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  np: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  sp: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  g: runtime: ?, size: EXP

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   fib(z) -{ 5 }-> s3 :|: s3 >= 0, s3 <= 1 + (1 + 0) + 0, z = 1 + (1 + 0)
   fib(z) -{ 6 }-> s4 :|: s4 >= 0, s4 <= 1 + (1 + 0) + (1 + 0), z = 1 + (1 + (1 + 0))
   fib(z) -{ 2 }-> sp(np(g(z - 3))) :|: z - 3 >= 0
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 5 }-> s1 :|: s1 >= 0, s1 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 6 }-> s2 :|: s2 >= 0, s2 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 2 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 2 + y }-> 1 + s' + x :|: s' >= 0, s' <= x + y, z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 1 + z' }-> 1 + s'' :|: s'' >= 0, s'' <= z + (z' - 1), z >= 0, z' - 1 >= 0
   sp(z) -{ 2 + y }-> s :|: s >= 0, s <= x + y, z = 1 + x + y, x >= 0, y >= 0

Function symbols to be analyzed: {g}, {fib}
Previous analysis results are:
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z + z']
  np: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  sp: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  g: runtime: INF, size: EXP

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

(45) 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:

   sp(v0) -> null_sp [0]
   np(v0) -> null_np [0]
   fib(v0) -> null_fib [0]
   g(v0) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

And the following fresh constants:    null_sp, null_np, null_fib, null_g, null_plus

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

(46)
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:

   fib(0) -> 0 [1]
   fib(s(0)) -> s(0) [1]
   fib(s(s(0))) -> s(0) [1]
   fib(s(s(x))) -> sp(g(x)) [1]
   g(0) -> pair(s(0), 0) [1]
   g(s(0)) -> pair(s(0), s(0)) [1]
   g(s(x)) -> np(g(x)) [1]
   sp(pair(x, y)) -> plus(x, y) [1]
   np(pair(x, y)) -> pair(plus(x, y), x) [1]
   plus(x, 0) -> x [1]
   plus(x, s(y)) -> s(plus(x, y)) [1]
   sp(v0) -> null_sp [0]
   np(v0) -> null_np [0]
   fib(v0) -> null_fib [0]
   g(v0) -> null_g [0]
   plus(v0, v1) -> null_plus [0]

The TRS has the following type information:
fib :: 0:s:null_sp:null_fib:null_plus -> 0:s:null_sp:null_fib:null_plus
0 :: 0:s:null_sp:null_fib:null_plus
s :: 0:s:null_sp:null_fib:null_plus -> 0:s:null_sp:null_fib:null_plus
sp :: pair:null_np:null_g -> 0:s:null_sp:null_fib:null_plus
g :: 0:s:null_sp:null_fib:null_plus -> pair:null_np:null_g
pair :: 0:s:null_sp:null_fib:null_plus -> 0:s:null_sp:null_fib:null_plus -> pair:null_np:null_g
np :: pair:null_np:null_g -> pair:null_np:null_g
plus :: 0:s:null_sp:null_fib:null_plus -> 0:s:null_sp:null_fib:null_plus -> 0:s:null_sp:null_fib:null_plus
null_sp :: 0:s:null_sp:null_fib:null_plus
null_np :: pair:null_np:null_g
null_fib :: 0:s:null_sp:null_fib:null_plus
null_g :: pair:null_np:null_g
null_plus :: 0:s:null_sp:null_fib:null_plus

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

(47) 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
   null_sp => 0
   null_np => 0
   null_fib => 0
   null_g => 0
   null_plus => 0

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

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

   fib(z) -{ 1 }-> sp(g(x)) :|: x >= 0, z = 1 + (1 + x)
   fib(z) -{ 1 }-> 0 :|: z = 0
   fib(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + 0
   fib(z) -{ 1 }-> 1 + 0 :|: z = 1 + (1 + 0)
   g(z) -{ 1 }-> np(g(x)) :|: x >= 0, z = 1 + x
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 1 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 1 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   np(z) -{ 1 }-> 1 + plus(x, y) + x :|: z = 1 + x + y, x >= 0, y >= 0
   plus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   sp(z) -{ 1 }-> plus(x, y) :|: z = 1 + x + y, x >= 0, y >= 0
   sp(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

Only complete derivations are relevant for the runtime complexity.

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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(0))) -> c2
   FIB(s(s(z0))) -> c3(SP(g(z0)), G(z0))
   G(0) -> c4
   G(s(0)) -> c5
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, 0) -> c9
   +'(z0, s(z1)) -> c10(+'(z0, z1))
S tuples:
   FIB(0) -> c
   FIB(s(0)) -> c1
   FIB(s(s(0))) -> c2
   FIB(s(s(z0))) -> c3(SP(g(z0)), G(z0))
   G(0) -> c4
   G(s(0)) -> c5
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, 0) -> c9
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   fib_1, g_1, sp_1, np_1, +_2

Defined Pair Symbols:   FIB_1, G_1, SP_1, NP_1, +'_2

Compound Symbols:   c, c1, c2, c3_2, c4, c5, c6_2, c7_1, c8_1, c9, c10_1


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

(51) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 6 trailing nodes:
   FIB(s(s(0))) -> c2
   FIB(s(0)) -> c1
   FIB(0) -> c
   G(0) -> c4
   +'(z0, 0) -> c9
   G(s(0)) -> c5

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   FIB(s(s(z0))) -> c3(SP(g(z0)), G(z0))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
S tuples:
   FIB(s(s(z0))) -> c3(SP(g(z0)), G(z0))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:none
Defined Rule Symbols:   fib_1, g_1, sp_1, np_1, +_2

Defined Pair Symbols:   FIB_1, G_1, SP_1, NP_1, +'_2

Compound Symbols:   c3_2, c6_2, c7_1, c8_1, c10_1


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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   FIB(s(s(z0))) -> c(G(z0))
S tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   FIB(s(s(z0))) -> c(G(z0))
K tuples:none
Defined Rule Symbols:   fib_1, g_1, sp_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(55) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(s(s(z0))) -> c(G(z0))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
S tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
K tuples:none
Defined Rule Symbols:   fib_1, g_1, sp_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(57) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
S tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
Defined Rule Symbols:   fib_1, g_1, sp_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(59) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fib(0) -> 0
   fib(s(0)) -> s(0)
   fib(s(s(0))) -> s(0)
   fib(s(s(z0))) -> sp(g(z0))
   sp(pair(z0, z1)) -> +(z0, z1)

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
S tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(61) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
We considered the (Usable) Rules:none
And the Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [2] + x_1 + [2]x_2^2 + [2]x_1^2
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FIB(x_1)) = 0
   POL(G(x_1)) = [2]x_1^2
   POL(NP(x_1)) = 0
   POL(SP(x_1)) = 0
   POL(c(x_1)) = x_1
   POL(c10(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(g(x_1)) = 0
   POL(np(x_1)) = [1]
   POL(pair(x_1, x_2)) = 0
   POL(s(x_1)) = [2] + x_1

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
S tuples:
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   NP(pair(z0, z1)) -> c8(+'(z0, z1))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, SP_1, NP_1, +'_2, FIB_1

Compound Symbols:   c6_2, c7_1, c8_1, c10_1, c_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(z0)) -> c6(NP(g(z0)), G(z0)) by 
   G(s(0)) -> c6(NP(pair(s(0), 0)), G(0))
   G(s(s(0))) -> c6(NP(pair(s(0), s(0))), G(s(0)))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)), G(0))
   G(s(s(0))) -> c6(NP(pair(s(0), s(0))), G(s(0)))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, FIB_1, G_1

Compound Symbols:   c7_1, c8_1, c10_1, c_1, c6_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   G(s(s(0))) -> c6(NP(pair(s(0), s(0))), G(s(0)))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, FIB_1, G_1

Compound Symbols:   c7_1, c8_1, c10_1, c_1, c6_2, c6_1


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, FIB_1, G_1

Compound Symbols:   c7_1, c8_1, c10_1, c_1, c6_2, c6_1, c1_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIB(s(s(z0))) -> c(SP(g(z0))) by 
   FIB(s(s(0))) -> c(SP(pair(s(0), 0)))
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))
   FIB(s(s(s(z0)))) -> c(SP(np(g(z0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   FIB(s(s(0))) -> c(SP(pair(s(0), 0)))
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))
   FIB(s(s(s(z0)))) -> c(SP(np(g(z0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   FIB(s(s(z0))) -> c(SP(g(z0)))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_2, c6_1, c1_1, c_1


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

(73) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   FIB(s(s(0))) -> c(SP(pair(s(0), 0)))
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   FIB(s(s(s(z0)))) -> c(SP(np(g(z0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_2, c6_1, c1_1, c_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c6(NP(np(g(z0))), G(s(z0))) by 
   G(s(s(0))) -> c6(NP(np(pair(s(0), 0))), G(s(0)))
   G(s(s(s(0)))) -> c6(NP(np(pair(s(0), s(0)))), G(s(s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   FIB(s(s(s(z0)))) -> c(SP(np(g(z0))))
   G(s(s(0))) -> c6(NP(np(pair(s(0), 0))), G(s(0)))
   G(s(s(s(0)))) -> c6(NP(np(pair(s(0), s(0)))), G(s(s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_1, c1_1, c_1, c6_2


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   FIB(s(s(s(z0)))) -> c(SP(np(g(z0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(NP(np(pair(s(0), 0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(NP(np(pair(s(0), s(0)))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_1, c1_1, c_1, c6_2, c2_1


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

(79) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIB(s(s(s(z0)))) -> c(SP(np(g(z0)))) by 
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(NP(np(pair(s(0), 0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(NP(np(pair(s(0), s(0)))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_1, c1_1, c6_2, c2_1, c_1


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

(81) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(0))) -> c2(NP(np(pair(s(0), 0)))) by G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
----------------------------------------

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(NP(np(pair(s(0), s(0)))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_1, c1_1, c6_2, c2_1, c_1


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

(83) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(s(0)))) -> c2(NP(np(pair(s(0), s(0))))) by G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
----------------------------------------

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   SP_1, NP_1, +'_2, G_1, FIB_1

Compound Symbols:   c7_1, c8_1, c10_1, c6_1, c1_1, c6_2, c2_1, c_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SP(pair(z0, z1)) -> c7(+'(z0, z1)) by 
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   NP_1, +'_2, G_1, FIB_1, SP_1

Compound Symbols:   c8_1, c10_1, c6_1, c1_1, c6_2, c2_1, c_1, c7_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace NP(pair(z0, z1)) -> c8(+'(z0, z1)) by 
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(0))) -> c1(G(s(0)))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(0))) -> c2(G(s(0)))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   +'_2, G_1, FIB_1, SP_1, NP_1

Compound Symbols:   c10_1, c6_1, c1_1, c6_2, c2_1, c_1, c7_1, c8_1


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

(89) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   G(s(s(0))) -> c2(G(s(0)))
   G(s(0)) -> c6(NP(pair(s(0), 0)))
   G(s(s(0))) -> c1(G(s(0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
S tuples:
   +'(z0, s(z1)) -> c10(+'(z0, z1))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   +'_2, G_1, FIB_1, SP_1, NP_1

Compound Symbols:   c10_1, c1_1, c6_2, c2_1, c_1, c7_1, c8_1


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

(91) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(z1)) -> c10(+'(z0, z1)) by 
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(93) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(s(s(s(0)))) -> c(SP(np(pair(s(0), 0)))) by FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0))))
----------------------------------------

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0)))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(95) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(s(s(s(s(0))))) -> c(SP(np(pair(s(0), s(0))))) by FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
----------------------------------------

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(97) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(0))) -> c2(NP(pair(+(s(0), 0), s(0)))) by G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
----------------------------------------

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(99) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0)))) by G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
----------------------------------------

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(101) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(s(s(s(0)))) -> c(SP(pair(+(s(0), 0), s(0)))) by FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))
----------------------------------------

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(103) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0))))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0))))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(105) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(s(s(s(s(0))))) -> c(SP(pair(+(s(0), s(0)), s(0)))) by FIB(s(s(s(s(0))))) -> c(SP(pair(s(+(s(0), 0)), s(0))))
----------------------------------------

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(pair(s(+(s(0), 0)), s(0))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c7_1, c8_1, c10_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1))) by 
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   FIB(s(s(s(s(0))))) -> c(SP(pair(s(+(s(0), 0)), s(0))))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, NP_1, +'_2, SP_1

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c8_1, c10_1, c7_1


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

(109) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FIB(s(s(s(s(0))))) -> c(SP(pair(s(+(s(0), 0)), s(0))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, NP_1, +'_2, SP_1

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c8_1, c10_1, c7_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1))) by 
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, +'_2, SP_1, NP_1

Compound Symbols:   c1_1, c6_2, c2_1, c_1, c10_1, c7_1, c8_1


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

(113) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   G(s(s(0))) -> c1(NP(pair(s(0), s(0))))
   G(s(s(s(0)))) -> c2(G(s(s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   G(s(s(0))) -> c2(NP(pair(s(0), s(0))))

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

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
K tuples:
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, +'_2, SP_1, NP_1

Compound Symbols:   c6_2, c_1, c10_1, c7_1, c8_1


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(115) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace +'(z0, s(s(y1))) -> c10(+'(z0, s(y1))) by 
   +'(z0, s(s(s(y1)))) -> c10(+'(z0, s(s(y1))))

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(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   g(0) -> pair(s(0), 0)
   g(s(0)) -> pair(s(0), s(0))
   g(s(z0)) -> np(g(z0))
   np(pair(z0, z1)) -> pair(+(z0, z1), z0)
   +(z0, 0) -> z0
   +(z0, s(z1)) -> s(+(z0, z1))
Tuples:
   G(s(s(s(z0)))) -> c6(NP(np(np(g(z0)))), G(s(s(z0))))
   FIB(s(s(s(s(z0))))) -> c(SP(np(np(g(z0)))))
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
   +'(z0, s(s(s(y1)))) -> c10(+'(z0, s(s(y1))))
S tuples:
   +'(z0, s(s(s(y1)))) -> c10(+'(z0, s(s(y1))))
K tuples:
   SP(pair(z0, s(s(y1)))) -> c7(+'(z0, s(s(y1))))
   NP(pair(z0, s(s(y1)))) -> c8(+'(z0, s(s(y1))))
Defined Rule Symbols:   g_1, np_1, +_2

Defined Pair Symbols:   G_1, FIB_1, SP_1, NP_1, +'_2

Compound Symbols:   c6_2, c_1, c7_1, c8_1, c10_1