KILLED
proof of input_JMNp3qudlv.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(6) CdtProblem
    (7) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxRelTRS
    (9) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxRelTRS
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) typed CpxTrs
    (13) OrderProof [LOWER BOUND(ID), 14 ms]
    (14) typed CpxTrs
    (15) RewriteLemmaProof [LOWER BOUND(ID), 1306 ms]
    (16) typed CpxTrs
    (17) RewriteLemmaProof [LOWER BOUND(ID), 28.9 s]
    (18) proven lower bound
    (19) LowerBoundPropagationProof [FINISHED, 0 ms]
    (20) BOUNDS(n^1, INF)
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (28) CdtProblem
    (29) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (32) CdtProblem
    (33) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CpxRelTRS
        (35) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
        (36) CpxTRS
        (37) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (38) CpxWeightedTrs
            (39) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
            (40) CpxWeightedTrs
            (41) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
            (42) CpxTypedWeightedTrs
            (43) CompletionProof [UPPER BOUND(ID), 0 ms]
            (44) CpxTypedWeightedCompleteTrs
                (45) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
                (46) CpxTypedWeightedCompleteTrs
                (47) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (48) CpxRNTS
                (49) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
                (50) CpxRNTS
                (51) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
                (52) CpxRNTS
                (53) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (54) CpxRNTS
                (55) IntTrsBoundProof [UPPER BOUND(ID), 179 ms]
                (56) CpxRNTS
                (57) IntTrsBoundProof [UPPER BOUND(ID), 72 ms]
                (58) CpxRNTS
                (59) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (60) CpxRNTS
                (61) IntTrsBoundProof [UPPER BOUND(ID), 331 ms]
                (62) CpxRNTS
                (63) IntTrsBoundProof [UPPER BOUND(ID), 73 ms]
                (64) CpxRNTS
                (65) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (66) CpxRNTS
                (67) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
                (68) CpxRNTS
                (69) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
                (70) CpxRNTS
                (71) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (72) CpxRNTS
                (73) IntTrsBoundProof [UPPER BOUND(ID), 39 ms]
                (74) CpxRNTS
                (75) IntTrsBoundProof [UPPER BOUND(ID), 24 ms]
                (76) CpxRNTS
                (77) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (78) CpxRNTS
                (79) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
                (80) CpxRNTS
                (81) IntTrsBoundProof [UPPER BOUND(ID), 4 ms]
                (82) CpxRNTS
                (83) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (84) CpxRNTS
                (85) IntTrsBoundProof [UPPER BOUND(ID), 1505 ms]
                (86) CpxRNTS
                (87) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
                (88) CpxRNTS
                (89) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
                (90) CpxRNTS
                (91) IntTrsBoundProof [UPPER BOUND(ID), 274 ms]
                (92) CpxRNTS
                (93) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
                (94) CpxRNTS
            (95) CompletionProof [UPPER BOUND(ID), 0 ms]
            (96) CpxTypedWeightedCompleteTrs
                (97) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
                (98) CpxRNTS
    (99) CdtRuleRemovalProof [UPPER BOUND(ADD(n^3)), 141 ms]
    (100) CdtProblem
        (101) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
        (102) CdtProblem
        (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (104) CdtProblem
        (105) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
        (106) CdtProblem
        (107) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (108) CdtProblem
        (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 2 ms]
        (110) CdtProblem
        (111) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (112) CdtProblem
        (113) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (114) CdtProblem
        (115) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
        (116) CdtProblem
        (117) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (118) CdtProblem
        (119) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (120) CdtProblem
        (121) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (122) CdtProblem
        (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (124) CdtProblem
        (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (126) CdtProblem
        (127) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (128) CdtProblem
        (129) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (130) CdtProblem
        (131) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (132) CdtProblem
        (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (134) CdtProblem
        (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (136) CdtProblem
        (137) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (138) CdtProblem
        (139) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (140) CdtProblem
        (141) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
        (142) CdtProblem
        (143) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (144) CdtProblem
        (145) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (146) CdtProblem
        (147) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (148) CdtProblem
        (149) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (150) CdtProblem
        (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
        (152) CdtProblem
        (153) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (154) CdtProblem
(155) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(156) CpxWeightedTrs
    (157) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CpxWeightedTrs
    (159) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CpxTypedWeightedTrs
    (161) CompletionProof [UPPER BOUND(ID), 0 ms]
    (162) CpxTypedWeightedCompleteTrs
        (163) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (164) CpxTypedWeightedCompleteTrs
        (165) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (166) CpxRNTS
        (167) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (168) CpxRNTS
        (169) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (170) CpxRNTS
        (171) ResultPropagationProof [UPPER BOUND(ID), 2 ms]
        (172) CpxRNTS
        (173) IntTrsBoundProof [UPPER BOUND(ID), 217 ms]
        (174) CpxRNTS
        (175) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
        (176) CpxRNTS
        (177) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (178) CpxRNTS
        (179) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
        (180) CpxRNTS
        (181) IntTrsBoundProof [UPPER BOUND(ID), 56 ms]
        (182) CpxRNTS
        (183) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (184) CpxRNTS
        (185) IntTrsBoundProof [UPPER BOUND(ID), 89 ms]
        (186) CpxRNTS
        (187) IntTrsBoundProof [UPPER BOUND(ID), 55 ms]
        (188) CpxRNTS
        (189) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (190) CpxRNTS
        (191) IntTrsBoundProof [UPPER BOUND(ID), 1504 ms]
        (192) CpxRNTS
        (193) IntTrsBoundProof [UPPER BOUND(ID), 94 ms]
        (194) CpxRNTS
    (195) CompletionProof [UPPER BOUND(ID), 0 ms]
    (196) CpxTypedWeightedCompleteTrs
        (197) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (198) CpxRNTS


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

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

(6)
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


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

(7) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(8)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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))

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

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

(10)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   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))

The (relative) TRS S consists of the following 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))

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

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(12)
Obligation:
Innermost TRS:
Rules:
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))
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))

Types:
FIB :: 0':s:c9:c10 -> c:c1:c2:c3
0' :: 0':s:c9:c10
c :: c:c1:c2:c3
s :: 0':s:c9:c10 -> 0':s:c9:c10
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c7 -> c4:c5:c6 -> c:c1:c2:c3
SP :: pair -> c7
g :: 0':s:c9:c10 -> pair
G :: 0':s:c9:c10 -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c8 -> c4:c5:c6 -> c4:c5:c6
NP :: pair -> c8
pair :: 0':s:c9:c10 -> 0':s:c9:c10 -> pair
c7 :: 0':s:c9:c10 -> c7
+' :: 0':s:c9:c10 -> 0':s:c9:c10 -> 0':s:c9:c10
c8 :: 0':s:c9:c10 -> c8
c9 :: 0':s:c9:c10
c10 :: 0':s:c9:c10 -> 0':s:c9:c10
fib :: 0':s:c9:c10 -> 0':s:c9:c10
sp :: pair -> 0':s:c9:c10
np :: pair -> pair
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s:c9:c102_11 :: 0':s:c9:c10
hole_c73_11 :: c7
hole_c4:c5:c64_11 :: c4:c5:c6
hole_pair5_11 :: pair
hole_c86_11 :: c8
gen_0':s:c9:c107_11 :: Nat -> 0':s:c9:c10
gen_c4:c5:c68_11 :: Nat -> c4:c5:c6

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

(13) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
g, G, +'

They will be analysed ascendingly in the following order:
g < G

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

(14)
Obligation:
Innermost TRS:
Rules:
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))
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))

Types:
FIB :: 0':s:c9:c10 -> c:c1:c2:c3
0' :: 0':s:c9:c10
c :: c:c1:c2:c3
s :: 0':s:c9:c10 -> 0':s:c9:c10
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c7 -> c4:c5:c6 -> c:c1:c2:c3
SP :: pair -> c7
g :: 0':s:c9:c10 -> pair
G :: 0':s:c9:c10 -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c8 -> c4:c5:c6 -> c4:c5:c6
NP :: pair -> c8
pair :: 0':s:c9:c10 -> 0':s:c9:c10 -> pair
c7 :: 0':s:c9:c10 -> c7
+' :: 0':s:c9:c10 -> 0':s:c9:c10 -> 0':s:c9:c10
c8 :: 0':s:c9:c10 -> c8
c9 :: 0':s:c9:c10
c10 :: 0':s:c9:c10 -> 0':s:c9:c10
fib :: 0':s:c9:c10 -> 0':s:c9:c10
sp :: pair -> 0':s:c9:c10
np :: pair -> pair
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s:c9:c102_11 :: 0':s:c9:c10
hole_c73_11 :: c7
hole_c4:c5:c64_11 :: c4:c5:c6
hole_pair5_11 :: pair
hole_c86_11 :: c8
gen_0':s:c9:c107_11 :: Nat -> 0':s:c9:c10
gen_c4:c5:c68_11 :: Nat -> c4:c5:c6


Generator Equations:
gen_0':s:c9:c107_11(0) <=> 0'
gen_0':s:c9:c107_11(+(x, 1)) <=> s(gen_0':s:c9:c107_11(x))
gen_c4:c5:c68_11(0) <=> c4
gen_c4:c5:c68_11(+(x, 1)) <=> c6(c8(0'), gen_c4:c5:c68_11(x))


The following defined symbols remain to be analysed:
g, G, +'

They will be analysed ascendingly in the following order:
g < G

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

(15) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
g(gen_0':s:c9:c107_11(+(1, n10_11))) -> *9_11, rt in Omega(0)

Induction Base:
g(gen_0':s:c9:c107_11(+(1, 0)))

Induction Step:
g(gen_0':s:c9:c107_11(+(1, +(n10_11, 1)))) ->_R^Omega(0)
np(g(gen_0':s:c9:c107_11(+(1, n10_11)))) ->_IH
np(*9_11)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(16)
Obligation:
Innermost TRS:
Rules:
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))
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))

Types:
FIB :: 0':s:c9:c10 -> c:c1:c2:c3
0' :: 0':s:c9:c10
c :: c:c1:c2:c3
s :: 0':s:c9:c10 -> 0':s:c9:c10
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c7 -> c4:c5:c6 -> c:c1:c2:c3
SP :: pair -> c7
g :: 0':s:c9:c10 -> pair
G :: 0':s:c9:c10 -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c8 -> c4:c5:c6 -> c4:c5:c6
NP :: pair -> c8
pair :: 0':s:c9:c10 -> 0':s:c9:c10 -> pair
c7 :: 0':s:c9:c10 -> c7
+' :: 0':s:c9:c10 -> 0':s:c9:c10 -> 0':s:c9:c10
c8 :: 0':s:c9:c10 -> c8
c9 :: 0':s:c9:c10
c10 :: 0':s:c9:c10 -> 0':s:c9:c10
fib :: 0':s:c9:c10 -> 0':s:c9:c10
sp :: pair -> 0':s:c9:c10
np :: pair -> pair
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s:c9:c102_11 :: 0':s:c9:c10
hole_c73_11 :: c7
hole_c4:c5:c64_11 :: c4:c5:c6
hole_pair5_11 :: pair
hole_c86_11 :: c8
gen_0':s:c9:c107_11 :: Nat -> 0':s:c9:c10
gen_c4:c5:c68_11 :: Nat -> c4:c5:c6


Lemmas:
g(gen_0':s:c9:c107_11(+(1, n10_11))) -> *9_11, rt in Omega(0)


Generator Equations:
gen_0':s:c9:c107_11(0) <=> 0'
gen_0':s:c9:c107_11(+(x, 1)) <=> s(gen_0':s:c9:c107_11(x))
gen_c4:c5:c68_11(0) <=> c4
gen_c4:c5:c68_11(+(x, 1)) <=> c6(c8(0'), gen_c4:c5:c68_11(x))


The following defined symbols remain to be analysed:
G, +'
----------------------------------------

(17) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
+'(gen_0':s:c9:c107_11(a), gen_0':s:c9:c107_11(+(1, n3411261_11))) -> *9_11, rt in Omega(n3411261_11)

Induction Base:
+'(gen_0':s:c9:c107_11(a), gen_0':s:c9:c107_11(+(1, 0)))

Induction Step:
+'(gen_0':s:c9:c107_11(a), gen_0':s:c9:c107_11(+(1, +(n3411261_11, 1)))) ->_R^Omega(1)
c10(+'(gen_0':s:c9:c107_11(a), gen_0':s:c9:c107_11(+(1, n3411261_11)))) ->_IH
c10(*9_11)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(18)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
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))
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))

Types:
FIB :: 0':s:c9:c10 -> c:c1:c2:c3
0' :: 0':s:c9:c10
c :: c:c1:c2:c3
s :: 0':s:c9:c10 -> 0':s:c9:c10
c1 :: c:c1:c2:c3
c2 :: c:c1:c2:c3
c3 :: c7 -> c4:c5:c6 -> c:c1:c2:c3
SP :: pair -> c7
g :: 0':s:c9:c10 -> pair
G :: 0':s:c9:c10 -> c4:c5:c6
c4 :: c4:c5:c6
c5 :: c4:c5:c6
c6 :: c8 -> c4:c5:c6 -> c4:c5:c6
NP :: pair -> c8
pair :: 0':s:c9:c10 -> 0':s:c9:c10 -> pair
c7 :: 0':s:c9:c10 -> c7
+' :: 0':s:c9:c10 -> 0':s:c9:c10 -> 0':s:c9:c10
c8 :: 0':s:c9:c10 -> c8
c9 :: 0':s:c9:c10
c10 :: 0':s:c9:c10 -> 0':s:c9:c10
fib :: 0':s:c9:c10 -> 0':s:c9:c10
sp :: pair -> 0':s:c9:c10
np :: pair -> pair
hole_c:c1:c2:c31_11 :: c:c1:c2:c3
hole_0':s:c9:c102_11 :: 0':s:c9:c10
hole_c73_11 :: c7
hole_c4:c5:c64_11 :: c4:c5:c6
hole_pair5_11 :: pair
hole_c86_11 :: c8
gen_0':s:c9:c107_11 :: Nat -> 0':s:c9:c10
gen_c4:c5:c68_11 :: Nat -> c4:c5:c6


Lemmas:
g(gen_0':s:c9:c107_11(+(1, n10_11))) -> *9_11, rt in Omega(0)


Generator Equations:
gen_0':s:c9:c107_11(0) <=> 0'
gen_0':s:c9:c107_11(+(x, 1)) <=> s(gen_0':s:c9:c107_11(x))
gen_c4:c5:c68_11(0) <=> c4
gen_c4:c5:c68_11(+(x, 1)) <=> c6(c8(0'), gen_c4:c5:c68_11(x))


The following defined symbols remain to be analysed:
+'
----------------------------------------

(19) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(20)
BOUNDS(n^1, INF)

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

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

(22)
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


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

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

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

(24)
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


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

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

(26)
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


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

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

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

(28)
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


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

(29) 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))

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

(30)
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


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

(31) 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)

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

(32)
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


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

(33) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(34)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))

The (relative) TRS S consists of the following rules:

   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   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))

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

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

(36)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   G(s(z0)) -> c6(NP(g(z0)), G(z0))
   NP(pair(z0, z1)) -> c8(+'(z0, z1))
   +'(z0, s(z1)) -> c10(+'(z0, z1))
   SP(pair(z0, z1)) -> c7(+'(z0, z1))
   FIB(s(s(z0))) -> c(SP(g(z0)))
   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))

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

(37) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(38)
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:

   G(s(z0)) -> c6(NP(g(z0)), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(z0))) -> c(SP(g(z0))) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(z0)) -> np(g(z0)) [0]
   np(pair(z0, z1)) -> pair(+(z0, z1), z0) [0]
   +(z0, 0) -> z0 [0]
   +(z0, s(z1)) -> s(+(z0, z1)) [0]

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

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

+ => plus

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

(40)
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:

   G(s(z0)) -> c6(NP(g(z0)), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(z0))) -> c(SP(g(z0))) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(z0)) -> np(g(z0)) [0]
   np(pair(z0, z1)) -> pair(plus(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]

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

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

(42)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   G(s(z0)) -> c6(NP(g(z0)), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(z0))) -> c(SP(g(z0))) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(z0)) -> np(g(z0)) [0]
   np(pair(z0, z1)) -> pair(plus(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]

The TRS has the following type information:
G :: s:0 -> c6
s :: s:0 -> s:0
c6 :: c8 -> c6 -> c6
NP :: pair -> c8
g :: s:0 -> pair
pair :: s:0 -> s:0 -> pair
c8 :: c10 -> c8
+' :: s:0 -> s:0 -> c10
c10 :: c10 -> c10
SP :: pair -> c7
c7 :: c10 -> c7
FIB :: s:0 -> c
c :: c7 -> c
0 :: s:0
np :: pair -> pair
plus :: s:0 -> s:0 -> s:0

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

(43) 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:

   G_1
   NP_1
   +'_2

(c) The following functions are completely defined:

   SP_1
   FIB_1
   g_1
   np_1
   plus_2

Due to the following rules being added:

   SP(v0) -> const4 [0]
   FIB(v0) -> const5 [0]
   g(v0) -> const2 [0]
   np(v0) -> const2 [0]
   plus(v0, v1) -> 0 [0]

And the following fresh constants:    const4, const5, const2, const, const1, const3

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

(44)
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:

   G(s(z0)) -> c6(NP(g(z0)), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(z0))) -> c(SP(g(z0))) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(z0)) -> np(g(z0)) [0]
   np(pair(z0, z1)) -> pair(plus(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]
   SP(v0) -> const4 [0]
   FIB(v0) -> const5 [0]
   g(v0) -> const2 [0]
   np(v0) -> const2 [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
G :: s:0 -> c6
s :: s:0 -> s:0
c6 :: c8 -> c6 -> c6
NP :: pair:const2 -> c8
g :: s:0 -> pair:const2
pair :: s:0 -> s:0 -> pair:const2
c8 :: c10 -> c8
+' :: s:0 -> s:0 -> c10
c10 :: c10 -> c10
SP :: pair:const2 -> c7:const4
c7 :: c10 -> c7:const4
FIB :: s:0 -> c:const5
c :: c7:const4 -> c:const5
0 :: s:0
np :: pair:const2 -> pair:const2
plus :: s:0 -> s:0 -> s:0
const4 :: c7:const4
const5 :: c:const5
const2 :: pair:const2
const :: c6
const1 :: c8
const3 :: c10

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

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

(46)
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:

   G(s(0)) -> c6(NP(pair(s(0), 0)), G(0)) [1]
   G(s(s(0))) -> c6(NP(pair(s(0), s(0))), G(s(0))) [1]
   G(s(s(z0'))) -> c6(NP(np(g(z0'))), G(s(z0'))) [1]
   G(s(z0)) -> c6(NP(const2), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(0))) -> c(SP(pair(s(0), 0))) [0]
   FIB(s(s(s(0)))) -> c(SP(pair(s(0), s(0)))) [0]
   FIB(s(s(s(z0'')))) -> c(SP(np(g(z0'')))) [0]
   FIB(s(s(z0))) -> c(SP(const2)) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(0)) -> np(pair(s(0), 0)) [0]
   g(s(s(0))) -> np(pair(s(0), s(0))) [0]
   g(s(s(z01))) -> np(np(g(z01))) [0]
   g(s(z0)) -> np(const2) [0]
   np(pair(z0, z1)) -> pair(plus(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]
   SP(v0) -> const4 [0]
   FIB(v0) -> const5 [0]
   g(v0) -> const2 [0]
   np(v0) -> const2 [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
G :: s:0 -> c6
s :: s:0 -> s:0
c6 :: c8 -> c6 -> c6
NP :: pair:const2 -> c8
g :: s:0 -> pair:const2
pair :: s:0 -> s:0 -> pair:const2
c8 :: c10 -> c8
+' :: s:0 -> s:0 -> c10
c10 :: c10 -> c10
SP :: pair:const2 -> c7:const4
c7 :: c10 -> c7:const4
FIB :: s:0 -> c:const5
c :: c7:const4 -> c:const5
0 :: s:0
np :: pair:const2 -> pair:const2
plus :: s:0 -> s:0 -> s:0
const4 :: c7:const4
const5 :: c:const5
const2 :: pair:const2
const :: c6
const1 :: c8
const3 :: c10

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
   const4 => 0
   const5 => 0
   const2 => 0
   const => 0
   const1 => 0
   const3 => 0

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   FIB(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z0''))) :|: z = 1 + (1 + (1 + z0'')), z0'' >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z0 >= 0, z = 1 + (1 + z0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z0'))) + G(1 + z0') :|: z = 1 + (1 + z0'), z0' >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z0) :|: z = 1 + z0, z0 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z01))) :|: z = 1 + (1 + z01), z01 >= 0
   g(z) -{ 0 }-> np(0) :|: z = 1 + z0, z0 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1


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

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0


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

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

   { +' }
   { plus }
   { SP }
   { NP }
   { np }
   { g }
   { FIB }
   { G }

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {+'}, {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}

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

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {+'}, {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}

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

(55) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: +'
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {+'}, {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: ?, size: O(1) [0]

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

(57) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +'(z, z') -{ 1 }-> 1 + +'(z, z' - 1) :|: z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]

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

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

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

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]

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

(61) 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'

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

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

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {plus}, {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: ?, size: O(n^1) [z + z']

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

(63) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: plus
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + plus(z, z' - 1) :|: z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

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

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

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

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

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

(67) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(68)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {SP}, {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: ?, size: O(1) [1]

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

(69) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   FIB(z) -{ 0 }-> 1 + SP(0) :|: z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + 0) :|: z = 1 + (1 + 0)
   FIB(z) -{ 0 }-> 1 + SP(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + (1 + 0))
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]

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

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

(72)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]

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

(73) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(74)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {NP}, {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: ?, size: O(1) [1]

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

(75) 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: z

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

(76)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   G(z) -{ 1 }-> 1 + NP(0) + G(z - 1) :|: z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + 0) + G(0) :|: z = 1 + 0
   G(z) -{ 1 }-> 1 + NP(1 + (1 + 0) + (1 + 0)) + G(1 + 0) :|: z = 1 + (1 + 0)
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]

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

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

(78)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]

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

(79) 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

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

(80)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {np}, {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: ?, size: O(n^1) [2*z]

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

(81) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: np
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(82)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> np(0) :|: z - 1 >= 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + 0) :|: z = 1 + 0
   g(z) -{ 0 }-> np(1 + (1 + 0) + (1 + 0)) :|: z = 1 + (1 + 0)
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]

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

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

(84)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]

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

(85) 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: ?

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

(86)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {g}, {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]
  g: runtime: ?, size: EXP

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

(87) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: g
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

(88)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   FIB(z) -{ 0 }-> 1 + SP(np(g(z - 3))) :|: z - 3 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   G(z) -{ 1 }-> 1 + NP(np(g(z - 2))) + G(1 + (z - 2)) :|: z - 2 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> np(np(g(z - 2))) :|: z - 2 >= 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]
  g: runtime: O(1) [0], size: EXP

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

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

(90)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ s16 }-> 1 + s17 :|: s15 >= 0, s15 <= inf', s16 >= 0, s16 <= 2 * s15, s17 >= 0, s17 <= 1, z - 3 >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   G(z) -{ 1 + s13 }-> 1 + s14 + G(1 + (z - 2)) :|: s12 >= 0, s12 <= inf, s13 >= 0, s13 <= 2 * s12, s14 >= 0, s14 <= 1, z - 2 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s20 :|: s18 >= 0, s18 <= inf'', s19 >= 0, s19 <= 2 * s18, s20 >= 0, s20 <= 2 * s19, z - 2 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]
  g: runtime: O(1) [0], size: EXP

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

(91) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(92)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ s16 }-> 1 + s17 :|: s15 >= 0, s15 <= inf', s16 >= 0, s16 <= 2 * s15, s17 >= 0, s17 <= 1, z - 3 >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   G(z) -{ 1 + s13 }-> 1 + s14 + G(1 + (z - 2)) :|: s12 >= 0, s12 <= inf, s13 >= 0, s13 <= 2 * s12, s14 >= 0, s14 <= 1, z - 2 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s20 :|: s18 >= 0, s18 <= inf'', s19 >= 0, s19 <= 2 * s18, s20 >= 0, s20 <= 2 * s19, z - 2 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]
  g: runtime: O(1) [0], size: EXP
  FIB: runtime: ?, size: O(1) [2]

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

(93) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(94)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ z' }-> 1 + s' :|: s' >= 0, s' <= 0, z' - 1 >= 0, z >= 0
   FIB(z) -{ 0 }-> 0 :|: z >= 0
   FIB(z) -{ s16 }-> 1 + s17 :|: s15 >= 0, s15 <= inf', s16 >= 0, s16 <= 2 * s15, s17 >= 0, s17 <= 1, z - 3 >= 0
   FIB(z) -{ 2 }-> 1 + s3 :|: s3 >= 0, s3 <= 1, z = 1 + (1 + 0)
   FIB(z) -{ 3 }-> 1 + s4 :|: s4 >= 0, s4 <= 1, z = 1 + (1 + (1 + 0))
   FIB(z) -{ 0 }-> 1 + s5 :|: s5 >= 0, s5 <= 1, z - 2 >= 0
   G(z) -{ 1 + s13 }-> 1 + s14 + G(1 + (z - 2)) :|: s12 >= 0, s12 <= inf, s13 >= 0, s13 <= 2 * s12, s14 >= 0, s14 <= 1, z - 2 >= 0
   G(z) -{ 3 }-> 1 + s6 + G(0) :|: s6 >= 0, s6 <= 1, z = 1 + 0
   G(z) -{ 4 }-> 1 + s7 + G(1 + 0) :|: s7 >= 0, s7 <= 1, z = 1 + (1 + 0)
   G(z) -{ 1 }-> 1 + s8 + G(z - 1) :|: s8 >= 0, s8 <= 1, z - 1 >= 0
   NP(z) -{ 1 + z1 }-> 1 + s :|: s >= 0, s <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: z >= 0
   SP(z) -{ z1 }-> 1 + s'' :|: s'' >= 0, s'' <= 0, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> s10 :|: s10 >= 0, s10 <= 2 * (1 + (1 + 0) + (1 + 0)), z = 1 + (1 + 0)
   g(z) -{ 0 }-> s11 :|: s11 >= 0, s11 <= 2 * 0, z - 1 >= 0
   g(z) -{ 0 }-> s20 :|: s18 >= 0, s18 <= inf'', s19 >= 0, s19 <= 2 * s18, s20 >= 0, s20 <= 2 * s19, z - 2 >= 0
   g(z) -{ 0 }-> s9 :|: s9 >= 0, s9 <= 2 * (1 + (1 + 0) + 0), z = 1 + 0
   g(z) -{ 0 }-> 0 :|: z >= 0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: z >= 0
   np(z) -{ 0 }-> 1 + s1 + z0 :|: s1 >= 0, s1 <= z0 + z1, z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + s2 :|: s2 >= 0, s2 <= z + (z' - 1), z' - 1 >= 0, z >= 0

Function symbols to be analyzed: {FIB}, {G}
Previous analysis results are:
  +': runtime: O(n^1) [z'], size: O(1) [0]
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  SP: runtime: O(n^1) [z], size: O(1) [1]
  NP: runtime: O(n^1) [z], size: O(1) [1]
  np: runtime: O(1) [0], size: O(n^1) [2*z]
  g: runtime: O(1) [0], size: EXP
  FIB: runtime: INF, size: O(1) [2]

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

(95) 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]
   FIB(v0) -> null_FIB [0]
   g(v0) -> null_g [0]
   np(v0) -> null_np [0]
   plus(v0, v1) -> null_plus [0]
   G(v0) -> null_G [0]
   NP(v0) -> null_NP [0]
   +'(v0, v1) -> null_+' [0]

And the following fresh constants:    null_SP, null_FIB, null_g, null_np, null_plus, null_G, null_NP, null_+'

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

(96)
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:

   G(s(z0)) -> c6(NP(g(z0)), G(z0)) [1]
   NP(pair(z0, z1)) -> c8(+'(z0, z1)) [1]
   +'(z0, s(z1)) -> c10(+'(z0, z1)) [1]
   SP(pair(z0, z1)) -> c7(+'(z0, z1)) [0]
   FIB(s(s(z0))) -> c(SP(g(z0))) [0]
   g(0) -> pair(s(0), 0) [0]
   g(s(0)) -> pair(s(0), s(0)) [0]
   g(s(z0)) -> np(g(z0)) [0]
   np(pair(z0, z1)) -> pair(plus(z0, z1), z0) [0]
   plus(z0, 0) -> z0 [0]
   plus(z0, s(z1)) -> s(plus(z0, z1)) [0]
   SP(v0) -> null_SP [0]
   FIB(v0) -> null_FIB [0]
   g(v0) -> null_g [0]
   np(v0) -> null_np [0]
   plus(v0, v1) -> null_plus [0]
   G(v0) -> null_G [0]
   NP(v0) -> null_NP [0]
   +'(v0, v1) -> null_+' [0]

The TRS has the following type information:
G :: s:0:null_plus -> c6:null_G
s :: s:0:null_plus -> s:0:null_plus
c6 :: c8:null_NP -> c6:null_G -> c6:null_G
NP :: pair:null_g:null_np -> c8:null_NP
g :: s:0:null_plus -> pair:null_g:null_np
pair :: s:0:null_plus -> s:0:null_plus -> pair:null_g:null_np
c8 :: c10:null_+' -> c8:null_NP
+' :: s:0:null_plus -> s:0:null_plus -> c10:null_+'
c10 :: c10:null_+' -> c10:null_+'
SP :: pair:null_g:null_np -> c7:null_SP
c7 :: c10:null_+' -> c7:null_SP
FIB :: s:0:null_plus -> c:null_FIB
c :: c7:null_SP -> c:null_FIB
0 :: s:0:null_plus
np :: pair:null_g:null_np -> pair:null_g:null_np
plus :: s:0:null_plus -> s:0:null_plus -> s:0:null_plus
null_SP :: c7:null_SP
null_FIB :: c:null_FIB
null_g :: pair:null_g:null_np
null_np :: pair:null_g:null_np
null_plus :: s:0:null_plus
null_G :: c6:null_G
null_NP :: c8:null_NP
null_+' :: c10:null_+'

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

(97) 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_FIB => 0
   null_g => 0
   null_np => 0
   null_plus => 0
   null_G => 0
   null_NP => 0
   null_+' => 0

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

(98)
Obligation:
Complexity RNTS consisting of the following rules:

   +'(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   +'(z, z') -{ 1 }-> 1 + +'(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1
   FIB(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   FIB(z) -{ 0 }-> 1 + SP(g(z0)) :|: z0 >= 0, z = 1 + (1 + z0)
   G(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   G(z) -{ 1 }-> 1 + NP(g(z0)) + G(z0) :|: z = 1 + z0, z0 >= 0
   NP(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   NP(z) -{ 1 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   SP(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   SP(z) -{ 0 }-> 1 + +'(z0, z1) :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   g(z) -{ 0 }-> np(g(z0)) :|: z = 1 + z0, z0 >= 0
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   g(z) -{ 0 }-> 1 + (1 + 0) + 0 :|: z = 0
   g(z) -{ 0 }-> 1 + (1 + 0) + (1 + 0) :|: z = 1 + 0
   np(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   np(z) -{ 0 }-> 1 + plus(z0, z1) + z0 :|: z1 >= 0, z0 >= 0, z = 1 + z0 + z1
   plus(z, z') -{ 0 }-> z0 :|: z = z0, z0 >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + plus(z0, z1) :|: z = z0, z1 >= 0, z0 >= 0, z' = 1 + z1

Only complete derivations are relevant for the runtime complexity.

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

(99) CdtRuleRemovalProof (UPPER BOUND(ADD(n^3)))
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)) = [1] + x_1 + x_2 + x_2^2 + x_1*x_2 + x_1^2 + x_1^3 + x_1^2*x_2 + x_1*x_2^2 + x_2^3
   POL(+'(x_1, x_2)) = 0
   POL(0) = 0
   POL(FIB(x_1)) = x_1 + x_1^3
   POL(G(x_1)) = x_1
   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)) = x_1^3
   POL(np(x_1)) = [1]
   POL(pair(x_1, x_2)) = 0
   POL(s(x_1)) = [1] + x_1

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

(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(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


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

(101) 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))

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

(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(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


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

(103) 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)))

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

(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:
   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


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

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

(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:
   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


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

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

(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:
   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


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

(109) 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))))

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

(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:
   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


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

(111) 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))))

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

(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:
   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


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

(113) 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))))

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

(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:
   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


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

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

(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:
   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


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

(117) 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)))))

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

(118)
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


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

(119) 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))))
----------------------------------------

(120)
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


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

(121) 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))))
----------------------------------------

(122)
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


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

(123) 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))))
----------------------------------------

(124)
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(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))))
   FIB(s(s(s(0)))) -> c(SP(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


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

(125) 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))))
----------------------------------------

(126)
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(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))))
   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(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


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

(127) 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))))
----------------------------------------

(128)
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(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   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(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


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

(129) 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)))

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

(130)
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(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   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))))
   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


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

(131) 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)))

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

(132)
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(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   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))))
   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


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

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

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

(134)
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(s(z0))))) -> c(SP(np(np(g(z0)))))
   G(s(s(s(0)))) -> c2(NP(pair(+(s(0), s(0)), s(0))))
   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))))
   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


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

(135) 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))))
----------------------------------------

(136)
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(s(z0))))) -> c(SP(np(np(g(z0)))))
   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))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
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


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

(137) 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)))

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

(138)
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)))))
   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))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(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


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

(139) 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))))
----------------------------------------

(140)
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)))))
   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))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   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


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

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

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

(142)
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)))))
   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))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(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


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

(143) 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))))
----------------------------------------

(144)
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), s(0))))
   SP(pair(z0, s(y1))) -> c7(+'(z0, s(y1)))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   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


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

(145) 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))))

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

(146)
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), s(0))))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   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


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

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

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

(148)
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), s(0))))
   NP(pair(z0, s(y1))) -> c8(+'(z0, s(y1)))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(z0, s(s(y1))) -> c10(+'(z0, s(y1)))
   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


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

(149) 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))))

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

(150)
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), s(0))))
   G(s(s(s(0)))) -> c2(NP(pair(s(+(s(0), 0)), s(0))))
   +'(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:   c1_1, c6_2, c2_1, c_1, c10_1, c7_1, c8_1


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

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

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

(152)
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


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

(153) 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))))

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

(154)
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


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

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

(156)
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
----------------------------------------

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

+ => plus

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

(158)
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
----------------------------------------

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

(160)
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
----------------------------------------

(161) 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

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

(162)
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
----------------------------------------

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

(164)
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
----------------------------------------

(165) 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

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

(166)
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


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

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

(168)
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


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

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

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

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

(170)
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}

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

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

(172)
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}

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

(173) 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'

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

(174)
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']

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

(175) 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'

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

(176)
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']

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

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

(178)
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']

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

(179) 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

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

(180)
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]

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

(181) 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

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

(182)
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]

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

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

(184)
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]

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

(185) 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

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

(186)
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]

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

(187) 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

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

(188)
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]

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

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

(190)
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]

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

(191) 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: ?

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

(192)
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

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

(193) 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: ?

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

(194)
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

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

(195) 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

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

(196)
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
----------------------------------------

(197) 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

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

(198)
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.