### (0) Obligation:

Runtime Complexity TRS:
The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0), s(0)))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0, cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0) → 0
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Rewrite Strategy: FULL

### (1) DecreasingLoopProof (EQUIVALENT transformation)

The following loop(s) give(s) rise to the lower bound Ω(2n):
The rewrite sequence
mark(fib(fib1(X141735_3, X241736_3))) →+ a__fib(cons(mark(mark(X141735_3)), fib1(mark(X241736_3), add(mark(X141735_3), mark(X241736_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,0,0].
The pumping substitution is [X141735_3 / fib(fib1(X141735_3, X241736_3))].
The result substitution is [ ].

The rewrite sequence
mark(fib(fib1(X141735_3, X241736_3))) →+ a__fib(cons(mark(mark(X141735_3)), fib1(mark(X241736_3), add(mark(X141735_3), mark(X241736_3)))))
gives rise to a decreasing loop by considering the right hand sides subterm at position [0,1,1,0].
The pumping substitution is [X141735_3 / fib(fib1(X141735_3, X241736_3))].
The result substitution is [ ].

### (3) RenamingProof (EQUIVALENT transformation)

Renamed function symbols to avoid clashes with predefined symbol.

### (4) Obligation:

Runtime Complexity Relative TRS:
The TRS R consists of the following rules:

a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

S is empty.
Rewrite Strategy: FULL

Infered types.

### (6) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

### (7) OrderProof (LOWER BOUND(ID) transformation)

Heuristically decided to analyse the following defined symbols:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (8) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (9) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel.

### (10) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (11) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:
0'

Induction Step:

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (13) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (14) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__fib.

### (15) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (16) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__fib1.

### (17) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (18) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:

Induction Step:

We have rt ∈ Ω(n2) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n2).

### (20) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__sel, a__fib, mark, a__fib1

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (21) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__sel.

### (22) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
mark, a__fib, a__fib1

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (23) RewriteLemmaProof (LOWER BOUND(ID) transformation)

Proved the following rewrite lemma:

Induction Base:
0'

Induction Step:

We have rt ∈ Ω(n1) and sz ∈ O(n). Thus, we have ircR ∈ Ω(n).

### (25) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__fib, a__fib1

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (26) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__fib.

### (27) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

The following defined symbols remain to be analysed:
a__fib1

They will be analysed ascendingly in the following order:
a__fib = a__sel
a__fib = mark
a__fib = a__fib1
a__sel = mark
a__sel = a__fib1
mark = a__fib1

### (28) NoRewriteLemmaProof (LOWER BOUND(ID) transformation)

Could not prove a rewrite lemma for the defined symbol a__fib1.

### (29) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (30) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (32) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (33) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (35) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas:

Generator Equations:

No more defined symbols left to analyse.

### (36) LowerBoundsProof (EQUIVALENT transformation)

The lowerbound Ω(n2) was proven with the following lemma:

### (38) Obligation:

TRS:
Rules:
a__fib(N) → a__sel(mark(N), a__fib1(s(0'), s(0')))
a__fib1(X, Y) → cons(mark(X), fib1(Y, add(X, Y)))
a__sel(0', cons(X, XS)) → mark(X)
a__sel(s(N), cons(X, XS)) → a__sel(mark(N), mark(XS))
mark(fib(X)) → a__fib(mark(X))
mark(sel(X1, X2)) → a__sel(mark(X1), mark(X2))
mark(fib1(X1, X2)) → a__fib1(mark(X1), mark(X2))
mark(s(X)) → s(mark(X))
mark(0') → 0'
mark(cons(X1, X2)) → cons(mark(X1), X2)
a__fib(X) → fib(X)
a__sel(X1, X2) → sel(X1, X2)
a__fib1(X1, X2) → fib1(X1, X2)

Types:

Lemmas: