MAYBE
proof of input_1qpsj9z7GO.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 1 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 183 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 48 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 64 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 105 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 66 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 258 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 58 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 571 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 53 ms]
        (48) CpxRNTS
    (49) CompletionProof [UPPER BOUND(ID), 0 ms]
    (50) CpxTypedWeightedCompleteTrs
        (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
(53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (60) CdtProblem
    (61) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 293 ms]
    (66) CdtProblem
    (67) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (88) CdtProblem
    (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (94) CdtProblem
    (95) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (102) CdtProblem
    (103) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem


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

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


The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

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(N) -> sel(N, fib1(s(0'), s(0')))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   add(0', X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0', cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

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(N) -> sel(N, fib1(s(0), s(0)))
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y)))
   add(0, X) -> X
   add(s(X), Y) -> s(add(X, Y))
   sel(0, cons(X, XS)) -> X
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS))
   fib1(X1, X2) -> n__fib1(X1, X2)
   activate(n__fib1(X1, X2)) -> fib1(X1, X2)
   activate(X) -> X

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

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

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


The TRS R consists of the following rules:

   fib(N) -> sel(N, fib1(s(0), s(0))) [1]
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   sel(0, cons(X, XS)) -> X [1]
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1]
   fib1(X1, X2) -> n__fib1(X1, X2) [1]
   activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1]
   activate(X) -> X [1]

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

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

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

   fib(N) -> sel(N, fib1(s(0), s(0))) [1]
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   sel(0, cons(X, XS)) -> X [1]
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1]
   fib1(X1, X2) -> n__fib1(X1, X2) [1]
   activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
fib :: 0:s -> 0:s
sel :: 0:s -> n__fib1:cons -> 0:s
fib1 :: 0:s -> 0:s -> n__fib1:cons
s :: 0:s -> 0:s
0 :: 0:s
cons :: 0:s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0:s -> 0:s -> n__fib1:cons
add :: 0:s -> 0:s -> 0:s
activate :: n__fib1:cons -> n__fib1:cons

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

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

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

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   fib_1
   sel_2

(c) The following functions are completely defined:

   fib1_2
   activate_1
   add_2

Due to the following rules being added:
none

And the following fresh constants:    const

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

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

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

   fib(N) -> sel(N, fib1(s(0), s(0))) [1]
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   sel(0, cons(X, XS)) -> X [1]
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1]
   fib1(X1, X2) -> n__fib1(X1, X2) [1]
   activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
fib :: 0:s -> 0:s
sel :: 0:s -> n__fib1:cons -> 0:s
fib1 :: 0:s -> 0:s -> n__fib1:cons
s :: 0:s -> 0:s
0 :: 0:s
cons :: 0:s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0:s -> 0:s -> n__fib1:cons
add :: 0:s -> 0:s -> 0:s
activate :: n__fib1:cons -> n__fib1:cons
const :: n__fib1:cons

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

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

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

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

   fib(N) -> sel(N, cons(s(0), n__fib1(s(0), add(s(0), s(0))))) [2]
   fib(N) -> sel(N, n__fib1(s(0), s(0))) [2]
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   sel(0, cons(X, XS)) -> X [1]
   sel(s(N), cons(X, n__fib1(X1', X2'))) -> sel(N, fib1(X1', X2')) [2]
   sel(s(N), cons(X, XS)) -> sel(N, XS) [2]
   fib1(X1, X2) -> n__fib1(X1, X2) [1]
   activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1]
   activate(X) -> X [1]

The TRS has the following type information:
fib :: 0:s -> 0:s
sel :: 0:s -> n__fib1:cons -> 0:s
fib1 :: 0:s -> 0:s -> n__fib1:cons
s :: 0:s -> 0:s
0 :: 0:s
cons :: 0:s -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0:s -> 0:s -> n__fib1:cons
add :: 0:s -> 0:s -> 0:s
activate :: n__fib1:cons -> n__fib1:cons
const :: n__fib1:cons

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

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

   0 => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   fib(z) -{ 2 }-> sel(N, 1 + (1 + 0) + (1 + 0)) :|: z = N, N >= 0
   fib(z) -{ 2 }-> sel(N, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z = N, N >= 0
   fib1(z, z') -{ 1 }-> 1 + X + (1 + Y + add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X
   fib1(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(N, XS) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
   sel(z, z') -{ 2 }-> sel(N, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, z = 1 + N, X >= 0, z' = 1 + X + (1 + X1' + X2'), N >= 0


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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0


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

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

   { add }
   { fib1 }
   { activate }
   { sel }
   { fib }

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib}

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {add}, {fib1}, {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: ?, size: O(n^1) [z + z']

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 }-> 1 + add(z - 1, z') :|: z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + add(1 + 0, 1 + 0))) :|: z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 1 }-> 1 + z + (1 + z' + add(z, z')) :|: z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: fib1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + 2*z + 2*z'

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {fib1}, {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: ?, size: O(n^1) [2 + 2*z + 2*z']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: fib1
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z

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

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

   activate(z) -{ 1 }-> z :|: z >= 0
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, fib1(X1', X2')) :|: X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {activate}, {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: ?, size: O(n^1) [2*z]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {sel}, {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]
  sel: runtime: ?, size: EXP

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 2 }-> sel(z, 1 + (1 + 0) + (1 + 0)) :|: z >= 0
   fib(z) -{ 4 }-> sel(z, 1 + (1 + 0) + (1 + (1 + 0) + s)) :|: s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 2 }-> sel(z - 1, XS) :|: z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0
   sel(z, z') -{ 4 + X1' }-> sel(z - 1, s1) :|: s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0

Function symbols to be analyzed: {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]
  sel: runtime: EXP, size: EXP

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0
   sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0

Function symbols to be analyzed: {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]
  sel: runtime: EXP, size: EXP

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0
   sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0

Function symbols to be analyzed: {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]
  sel: runtime: EXP, size: EXP
  fib: runtime: ?, size: INF

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

(47) 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: ?

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

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

   activate(z) -{ 3 + X1 }-> s2 :|: s2 >= 0, s2 <= 2 * X1 + 2 * X2 + 2, X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   activate(z) -{ 1 }-> z :|: z >= 0
   add(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   add(z, z') -{ 1 + z }-> 1 + s'' :|: s'' >= 0, s'' <= z - 1 + z', z' >= 0, z - 1 >= 0
   fib(z) -{ 4 + inf }-> s3 :|: s3 >= 0, s3 <= inf', s >= 0, s <= 1 + 0 + (1 + 0), z >= 0
   fib(z) -{ 2 + inf'' }-> s4 :|: s4 >= 0, s4 <= inf1, z >= 0
   fib1(z, z') -{ 1 }-> 1 + z + z' :|: z >= 0, z' >= 0
   fib1(z, z') -{ 2 + z }-> 1 + z + (1 + z' + s') :|: s' >= 0, s' <= z + z', z' >= 0, z >= 0
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 4 + X1' + inf2 }-> s5 :|: s5 >= 0, s5 <= inf3, s1 >= 0, s1 <= 2 * X1' + 2 * X2' + 2, X2' >= 0, X1' >= 0, X >= 0, z' = 1 + X + (1 + X1' + X2'), z - 1 >= 0
   sel(z, z') -{ 2 + inf4 }-> s6 :|: s6 >= 0, s6 <= inf5, z' = 1 + X + XS, X >= 0, XS >= 0, z - 1 >= 0

Function symbols to be analyzed: {fib}
Previous analysis results are:
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: O(n^1) [2 + z], size: O(n^1) [2 + 2*z + 2*z']
  activate: runtime: O(n^1) [2 + z], size: O(n^1) [2*z]
  sel: runtime: EXP, size: EXP
  fib: runtime: INF, size: INF

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

(49) 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:

   sel(v0, v1) -> null_sel [0]
   add(v0, v1) -> null_add [0]

And the following fresh constants:    null_sel, null_add, const

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

(50)
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(N) -> sel(N, fib1(s(0), s(0))) [1]
   fib1(X, Y) -> cons(X, n__fib1(Y, add(X, Y))) [1]
   add(0, X) -> X [1]
   add(s(X), Y) -> s(add(X, Y)) [1]
   sel(0, cons(X, XS)) -> X [1]
   sel(s(N), cons(X, XS)) -> sel(N, activate(XS)) [1]
   fib1(X1, X2) -> n__fib1(X1, X2) [1]
   activate(n__fib1(X1, X2)) -> fib1(X1, X2) [1]
   activate(X) -> X [1]
   sel(v0, v1) -> null_sel [0]
   add(v0, v1) -> null_add [0]

The TRS has the following type information:
fib :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add
sel :: 0:s:null_sel:null_add -> n__fib1:cons -> 0:s:null_sel:null_add
fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> n__fib1:cons
s :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add
0 :: 0:s:null_sel:null_add
cons :: 0:s:null_sel:null_add -> n__fib1:cons -> n__fib1:cons
n__fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> n__fib1:cons
add :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 0:s:null_sel:null_add
activate :: n__fib1:cons -> n__fib1:cons
null_sel :: 0:s:null_sel:null_add
null_add :: 0:s:null_sel:null_add
const :: n__fib1:cons

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

(51) 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_sel => 0
   null_add => 0
   const => 0

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

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

   activate(z) -{ 1 }-> X :|: X >= 0, z = X
   activate(z) -{ 1 }-> fib1(X1, X2) :|: X1 >= 0, X2 >= 0, z = 1 + X1 + X2
   add(z, z') -{ 1 }-> X :|: z' = X, X >= 0, z = 0
   add(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   add(z, z') -{ 1 }-> 1 + add(X, Y) :|: z = 1 + X, z' = Y, Y >= 0, X >= 0
   fib(z) -{ 1 }-> sel(N, fib1(1 + 0, 1 + 0)) :|: z = N, N >= 0
   fib1(z, z') -{ 1 }-> 1 + X + (1 + Y + add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X
   fib1(z, z') -{ 1 }-> 1 + X1 + X2 :|: X1 >= 0, X2 >= 0, z = X1, z' = X2
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 1 }-> sel(N, activate(XS)) :|: z = 1 + N, z' = 1 + X + XS, X >= 0, XS >= 0, N >= 0
   sel(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
Tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   FIB1(z0, z1) -> c2
   ADD(0, z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   ACTIVATE(z0) -> c8
S tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   FIB1(z0, z1) -> c2
   ADD(0, z0) -> c3
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(0, cons(z0, z1)) -> c5
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   ACTIVATE(z0) -> c8
K tuples:none
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2, activate_1

Defined Pair Symbols:   FIB_1, FIB1_2, ADD_2, SEL_2, ACTIVATE_1

Compound Symbols:   c_2, c1_1, c2, c3, c4_1, c5, c6_2, c7_1, c8


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   FIB1(z0, z1) -> c2
   ACTIVATE(z0) -> c8
   SEL(0, cons(z0, z1)) -> c5
   ADD(0, z0) -> c3

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
Tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
S tuples:
   FIB(z0) -> c(SEL(z0, fib1(s(0), s(0))), FIB1(s(0), s(0)))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
K tuples:none
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2, activate_1

Defined Pair Symbols:   FIB_1, FIB1_2, ADD_2, SEL_2, ACTIVATE_1

Compound Symbols:   c_2, c1_1, c4_1, c6_2, c7_1


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   FIB(z0) -> c2(FIB1(s(0), s(0)))
S tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   FIB(z0) -> c2(FIB1(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2, activate_1

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(59) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(z0) -> c2(FIB1(s(0), s(0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
K tuples:none
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2, activate_1

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(61) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2, activate_1

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(63) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   sel(0, cons(z0, z1)) -> z0
   sel(s(z0), cons(z1, z2)) -> sel(z0, activate(z2))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
Defined Rule Symbols:   activate_1, fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(ACTIVATE(x_1)) = [1]
   POL(ADD(x_1, x_2)) = 0
   POL(FIB(x_1)) = [2] + [2]x_1 + [2]x_1^2
   POL(FIB1(x_1, x_2)) = 0
   POL(SEL(x_1, x_2)) = [2]x_1 + [2]x_1^2
   POL(activate(x_1)) = 0
   POL(add(x_1, x_2)) = [1] + [2]x_1 + [2]x_2 + [2]x_2^2 + [2]x_1*x_2 + [2]x_1^2
   POL(c1(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c4(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c7(x_1)) = x_1
   POL(cons(x_1, x_2)) = x_1
   POL(fib1(x_1, x_2)) = 0
   POL(n__fib1(x_1, x_2)) = 0
   POL(s(x_1)) = [1] + x_1

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
Defined Rule Symbols:   activate_1, fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(67) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   FIB1(z0, z1) -> c1(ADD(z0, z1))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   activate_1, fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, ACTIVATE_1, FIB_1

Compound Symbols:   c1_1, c4_1, c6_2, c7_1, c2_1


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2)) by 
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   activate_1, fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, FIB_1, SEL_2

Compound Symbols:   c1_1, c4_1, c7_1, c2_1, c6_2


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

(71) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   activate(n__fib1(z0, z1)) -> fib1(z0, z1)
   activate(z0) -> z0

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, FIB_1, SEL_2

Compound Symbols:   c1_1, c4_1, c7_1, c2_1, c6_2


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

(73) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0)))) by 
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
   FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
   FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0))))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c6(SEL(z0, activate(z2)), ACTIVATE(z2))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c2_1


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

(75) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   FIB(x0) -> c2(SEL(x0, n__fib1(s(0), s(0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c2_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1))) by 
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, n__fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, n__fib1(z0, z1)), ACTIVATE(n__fib1(z0, z1)))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c2_1


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   fib1_2, add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c2_1, c6_1


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

(81) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   fib1(z0, z1) -> cons(z0, n__fib1(z1, add(z0, z1)))
   fib1(z0, z1) -> n__fib1(z0, z1)

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0))))))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c2_1, c6_1


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

(83) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), add(s(0), s(0)))))) by 
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0)))))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0)))))))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c6_1, c2_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(add(0, s(0))))))) by 
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0))))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0))))))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2, FIB_1

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c6_1, c2_1


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

(87) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   FIB(x0) -> c2(SEL(x0, cons(s(0), n__fib1(s(0), s(s(0))))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   FIB1(z0, z1) -> c1(ADD(z0, z1))
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(z0, z1) -> c1(ADD(z0, z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   FIB1_2, ADD_2, ACTIVATE_1, SEL_2

Compound Symbols:   c1_1, c4_1, c7_1, c6_2, c6_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIB1(z0, z1) -> c1(ADD(z0, z1)) by 
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
S tuples:
   ADD(s(z0), z1) -> c4(ADD(z0, z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   ADD_2, ACTIVATE_1, SEL_2, FIB1_2

Compound Symbols:   c4_1, c7_1, c6_2, c6_1, c1_1


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

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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   ACTIVATE_1, SEL_2, FIB1_2, ADD_2

Compound Symbols:   c7_1, c6_2, c6_1, c1_1, c4_1


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

(93) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__fib1(z0, z1)) -> c7(FIB1(z0, z1)) by 
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

(95) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(x0), cons(x1, z0)) -> c6(SEL(x0, z0), ACTIVATE(z0)) by 
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))), ACTIVATE(cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(SEL(z0, n__fib1(s(y0), y1)), ACTIVATE(n__fib1(s(y0), y1)))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)), ACTIVATE(cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))), ACTIVATE(cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(SEL(z0, n__fib1(s(y0), y1)), ACTIVATE(n__fib1(s(y0), y1)))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

(99) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(ACTIVATE(n__fib1(z0, z1))) by 
   SEL(s(z0), cons(z1, n__fib1(s(y0), z3))) -> c6(ACTIVATE(n__fib1(s(y0), z3)))

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   FIB1(s(y0), z1) -> c1(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c1_1, c4_1, c7_1, c6_1


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

(101) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace FIB1(s(y0), z1) -> c1(ADD(s(y0), z1)) by 
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
S tuples:
   ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1))
K tuples:
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, ADD_2, ACTIVATE_1, FIB1_2

Compound Symbols:   c6_2, c4_1, c7_1, c6_1, c1_1


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

(103) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD(s(s(y0)), z1) -> c4(ADD(s(y0), z1)) by 
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
S tuples:
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
K tuples:
   ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, ACTIVATE_1, FIB1_2, ADD_2

Compound Symbols:   c6_2, c7_1, c6_1, c1_1, c4_1


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

(105) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ACTIVATE(n__fib1(s(y0), z1)) -> c7(FIB1(s(y0), z1)) by 
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
S tuples:
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
K tuples:
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

(107) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c6(SEL(s(y0), cons(y1, y2))) by 
   SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(y2, y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(y2, y3))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4)))))
   SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3))))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4)))))
   SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3))))
S tuples:
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
K tuples:
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

(109) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1))) by 
   SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3)))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   SEL(s(z0), cons(z1, n__fib1(s(y0), y1))) -> c6(ACTIVATE(n__fib1(s(y0), y1)))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4)))))
   SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3))))
   SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3)))
S tuples:
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
K tuples:
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1


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

(111) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(z0), cons(z1, n__fib1(s(y0), z3))) -> c6(ACTIVATE(n__fib1(s(y0), z3))) by 
   SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3)))

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(s(z0), z1) -> s(add(z0, z1))
   add(0, z0) -> z0
Tuples:
   SEL(s(x0), cons(x1, n__fib1(z0, z1))) -> c6(SEL(x0, cons(z0, n__fib1(z1, add(z0, z1)))), ACTIVATE(n__fib1(z0, z1)))
   SEL(s(s(y0)), cons(z1, cons(y1, n__fib1(y2, y3)))) -> c6(SEL(s(y0), cons(y1, n__fib1(y2, y3))))
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, n__fib1(y3, y4))))) -> c6(SEL(s(s(y0)), cons(z2, cons(y2, n__fib1(y3, y4)))))
   SEL(s(s(z0)), cons(z1, cons(z2, n__fib1(s(y2), y3)))) -> c6(SEL(s(z0), cons(z2, n__fib1(s(y2), y3))))
   SEL(s(z0), cons(z1, n__fib1(s(s(y0)), z3))) -> c6(ACTIVATE(n__fib1(s(s(y0)), z3)))
S tuples:
   ADD(s(s(s(y0))), z1) -> c4(ADD(s(s(y0)), z1))
K tuples:
   FIB1(s(s(y0)), z1) -> c1(ADD(s(s(y0)), z1))
   ACTIVATE(n__fib1(s(s(y0)), z1)) -> c7(FIB1(s(s(y0)), z1))
Defined Rule Symbols:   add_2

Defined Pair Symbols:   SEL_2, FIB1_2, ADD_2, ACTIVATE_1

Compound Symbols:   c6_2, c6_1, c1_1, c4_1, c7_1