KILLED
proof of input_VfeNxaapBw.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 413 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 61 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 293 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 26 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 5766 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 2036 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (48) CdtProblem
    (49) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 86 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) 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, 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, XS)

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, 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, XS)

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, 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, XS)

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, 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, XS) [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, 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, XS) [1]

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

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

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

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

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

   fib_1
   sel_2

(c) The following functions are completely defined:

   fib1_2
   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, 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, XS) [1]

The TRS has the following type information:
fib :: 0:s -> 0:s
sel :: 0:s -> cons -> 0:s
fib1 :: 0:s -> 0:s -> cons
s :: 0:s -> 0:s
0 :: 0:s
cons :: 0:s -> cons -> cons
add :: 0:s -> 0:s -> 0:s
const :: 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), fib1(s(0), add(s(0), s(0))))) [2]
   fib1(0, Y) -> cons(0, fib1(Y, Y)) [2]
   fib1(s(X'), Y) -> cons(s(X'), fib1(Y, s(add(X', Y)))) [2]
   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, XS) [1]

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

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

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


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

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

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

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

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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: sel
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:

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

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

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

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

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

(27) 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'

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

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

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

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

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

(29) 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

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

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

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

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

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

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

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

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

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: fib1
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

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

Function symbols to be analyzed: {fib1}, {fib}
Previous analysis results are:
  sel: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: ?, size: O(1) [0]

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: fib1
after applying outer abstraction to obtain an ITS,
resulting in: INF with polynomial bound: ?

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

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

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

Function symbols to be analyzed: {fib1}, {fib}
Previous analysis results are:
  sel: runtime: O(n^1) [1 + z], size: O(n^1) [z']
  add: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  fib1: runtime: INF, size: O(1) [0]

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

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

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

And the following fresh constants:    null_sel, null_add, const

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

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

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

   fib(N) -> sel(N, fib1(s(0), s(0))) [1]
   fib1(X, Y) -> cons(X, 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, XS) [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 -> cons -> 0:s:null_sel:null_add
fib1 :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 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 -> cons -> cons
add :: 0:s:null_sel:null_add -> 0:s:null_sel:null_add -> 0:s:null_sel:null_add
null_sel :: 0:s:null_sel:null_add
null_add :: 0:s:null_sel:null_add
const :: cons

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

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

   0 => 0
   null_sel => 0
   null_add => 0
   const => 0

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

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

   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 + fib1(Y, add(X, Y)) :|: z' = Y, Y >= 0, X >= 0, z = X
   sel(z, z') -{ 1 }-> X :|: z' = 1 + X + XS, X >= 0, z = 0, XS >= 0
   sel(z, z') -{ 1 }-> sel(N, 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.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   FIB_1, FIB1_2, ADD_2, SEL_2

Compound Symbols:   c_2, c1_2, c2, c3_1, c4, c5_1


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

(43) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   ADD(0, z0) -> c2
   SEL(0, cons(z0, z1)) -> c4

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

(44)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   FIB_1, FIB1_2, ADD_2, SEL_2

Compound Symbols:   c_2, c1_2, c3_1, c5_1


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

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2)
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   FIB(z0) -> c2(FIB1(s(0), s(0)))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   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

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

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

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

(48)
Obligation:
Complexity Dependency Tuples Problem

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

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(49) 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))))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   fib(z0) -> sel(z0, fib1(s(0), s(0)))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(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, z2)
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
Defined Rule Symbols:   fib_1, fib1_2, add_2, sel_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(51) 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, z2)

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
We considered the (Usable) Rules:none
And the Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [3]
   POL(ADD(x_1, x_2)) = 0
   POL(FIB(x_1)) = x_1
   POL(FIB1(x_1, x_2)) = 0
   POL(SEL(x_1, x_2)) = x_1
   POL(add(x_1, x_2)) = [1] + [3]x_1 + [2]x_2
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c5(x_1)) = x_1
   POL(cons(x_1, x_2)) = [3] + x_1
   POL(fib1(x_1, x_2)) = 0
   POL(s(x_1)) = [3] + x_1

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(55) 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), fib1(s(0), add(s(0), s(0))))))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), add(s(0), s(0))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(x0) -> c2(SEL(x0, cons(s(0), cons(s(0), fib1(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))
   FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0)))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(59) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(x0) -> c2(SEL(x0, cons(s(0), cons(s(0), fib1(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))
----------------------------------------

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0)))))))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(61) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(x0) -> c2(SEL(x0, cons(s(0), fib1(s(0), s(add(0, s(0))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))
----------------------------------------

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(z0), z1) -> c3(ADD(z0, z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, SEL_2, FIB_1

Compound Symbols:   c1_2, c3_1, c5_1, c2_1


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

(63) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD(s(z0), z1) -> c3(ADD(z0, z1)) by 
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, SEL_2, FIB_1, ADD_2

Compound Symbols:   c1_2, c5_1, c2_1, c3_1


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

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), fib1(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, SEL_2, FIB_1, ADD_2

Compound Symbols:   c1_2, c5_1, c2_1, c3_1


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

(67) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(z0), cons(z1, z2)) -> c5(SEL(z0, z2)) by 
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, FIB_1, ADD_2, SEL_2

Compound Symbols:   c1_2, c2_1, c3_1, c5_1


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), fib1(s(add(0, s(0))), add(s(0), s(add(0, s(0))))))))) by FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, ADD_2, FIB_1, SEL_2

Compound Symbols:   c1_2, c3_1, c2_1, c5_1


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD(s(s(y0)), z1) -> c3(ADD(s(y0), z1)) by 
   ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))))
   ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2)))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, FIB_1, SEL_2, ADD_2

Compound Symbols:   c1_2, c2_1, c5_1, c3_1


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

(73) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace SEL(s(s(y0)), cons(z1, cons(y1, y2))) -> c5(SEL(s(y0), cons(y1, y2))) by 
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3))))

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   add(0, z0) -> z0
   add(s(z0), z1) -> s(add(z0, z1))
   fib1(z0, z1) -> cons(z0, fib1(z1, add(z0, z1)))
Tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(add(s(0), s(0)), cons(add(s(0), add(s(0), s(0))), fib1(add(add(s(0), s(0)), add(s(0), add(s(0), s(0)))), add(add(s(0), add(s(0), s(0))), add(add(s(0), s(0)), add(s(0), add(s(0), s(0))))))))))))
   FIB(z0) -> c2(SEL(z0, cons(s(0), cons(s(0), cons(s(add(0, s(0))), fib1(add(s(0), s(add(0, s(0)))), add(s(add(0, s(0))), add(s(0), s(add(0, s(0)))))))))))
   ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
S tuples:
   FIB1(z0, z1) -> c1(FIB1(z1, add(z0, z1)), ADD(z0, z1))
   ADD(s(s(s(y0))), z1) -> c3(ADD(s(s(y0)), z1))
K tuples:
   FIB(z0) -> c2(SEL(z0, fib1(s(0), s(0))))
   SEL(s(s(s(y0))), cons(z1, cons(z2, cons(y2, y3)))) -> c5(SEL(s(s(y0)), cons(z2, cons(y2, y3))))
Defined Rule Symbols:   add_2, fib1_2

Defined Pair Symbols:   FIB1_2, FIB_1, ADD_2, SEL_2

Compound Symbols:   c1_2, c2_1, c3_1, c5_1